MSU LIBRARIES “ RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped he10w. _-.. .__1.__..___._—. PION INELASTIC SCATTERING AND THE PION-NUCLEUS EFFECTIVE INTERACTION By James Arthur Carr A DISSERTATION Submitted to Michigan State university in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1981 ABSTRACT PION INELASTIC SCATTERING AND THE PION-NUCLEUS EFFECTIVE INTERACTION By James Arthur Carr This work examines pion inelastic scattering with the primary purpose of gaining a better understanding of the properties of the pion-nucleus interaction. The main conclusion of the work is that an effective interaction which incorporates the most obvious theoretical corrections to the impulse approximation does a good job of explaining pion elastic and inelastic scattering from zero to 200 MeV without significant adjustments to the strength parameters of the force. Watson's multiple scattering theory is used to develop a theoret- ical interaction starting from the free pion-nucleon interaction. Elas- tic scattering was used to calibrate the isoscalar central interaction. It was found that the impulse approximation did poorly at low energy, while the multiple scattering corrections gave good agreement with all of the data after a few minor adjustments in the force. The distorted wave approximation for the inelastic transition matrix elements are evaluated for both natural and unnatural parity excitations. The isoscalar natural parity transitions are used to test the reaction theory, and it is found that the effective inter- action calibrated by elastic scattering produces good agreement with the inelastic data. Thus the medium corrections required to obtain the correct Optical potential are just as important when calculating James Arthur Carr inelastic cross sections. It is also noted that low energy inelastic scattering is much more sensitive to the choice of the distorting potential than is inelastic scattering in the resonance region. Calculations are also shown for other inelastic and charge exchange reactions. It appears that the isovector central inter- action is reasonable, but the importance of medium corrections cannot be determined. The unnatural parity transitions are also reasonably described by the theoretical estimate of the spin-orbit interaction, but not enough systematic data exists to reach a firm conclusion. In summary, it is seen that a consistent explanation of pion inelastic scattering is possible. In some cases more complete ex- perimental information is needed to prOperly test the theory. Other areas where more theoretical work is needed have also been identified. ACKNOWLEDGMENTS I would like to thank Prof. Hugh McManus for suggesting this problem, and for his patience and guidance throughout my graduate career. Our many discussions did much to deve10p my physical intui- tion and I shall always treasure the Opportunity I had to work with him. Thanks is also due to Dr. Karen Stricker, whose work on elastic scattering was an indispensible prerequisite to this project, Prof. Dan Olaf Riska for many useful discussions, and Prof. Fred Petrovich for invaluable insights and a very productive collaboration in the last year of this work. I would also like to thank the Cyclotron faculty, computer staff and Nuclear Beer Group for providing a stimu- lating work environment, and many experimental groups for supplying their data in advance of publication. Finally, special thanks to some people who mean a great deal to me. Joe Finck and I spent many years studying, teaching physics and partying together -- those years were some of the best of my life. (They were certainly unforgettable.) My parents and grandparents have always supported and encouraged me - I cannot thank them enough. Kathleen McGleish, now my wife, gave me her unwavering love and helped me through the long trying days and years it took to finish this. She has made my life full and complete. Words cannot express my gratitude. Last but not least, a big thank you to my mother-in- 1aw, Kelley, for the considerable effort involved in typing all of these equations! ii TABLE OF CONTENTS List of Figures List of Tables Chapter 1. 2. 6. INTRODUCTION THE PION-NUCLEUS INTERACTION 1 Multiple Scattering Theory 2 The First Order Potential 3 Calculations with First Order Optical Potential 4 Higher Order Corrections to the Pion Potential 5 Calculations with Full Optical Potential 6 Summary of Elastic Scattering Results INELASTIC SCATTERING FORMALISM 1 Overview of Scattering Theory and the DWBA 2 Reduction of the Cross Section Formula .3 Expansion of the Transition Matrix Elements .4 Evaluation of the Form Factor for Natural Parity Transitions 3.5 Evaluation of the Transition Matrix for Unnatural Parity Transition 3.6 Summary INELASTIC SCATTERING IN THE COLLECTIVE MODEL 1 Example of Collective Model at Low Energy 2 Example of Collective Model Near the Resonance 3 Distorted wave Effects 4 Other Low Energy Cases 5 Other High Energy Cases 6 Summary MICROSCOPIC MODELS FOR INELASTIC TRANSITIONS 5.1 Comparison of Collective and MicroscOpic Models 2 Other Natural Parity Cases 3 Unnatural Parity Transitions 4 Other Unnatural Parity Cases 5 Summary CHARGE EXCHANGE SCATTERING 6.1 Model for Charge Exchange Calculations 6.2 Sample Calculations 6.3 Summary iii Page ix 12 19 29 36 45 48 49 52 59 63 7O 72 74 75 81 87 104 125 137 139 139 152 162 166 174 176 176 179 185 Chapter 7. CONCLUSIONS Appendix A. PION SCATTERING AMPLITUDE B. EQUIVALENT FORMS OF THE POTENTIAL C. DENSITY PARAMETERS + + D. EVALUATE V°V TERM IN FORM FACTOR E. FOLDING MODEL FORMULAE REFERENCES iv 193 195 198 200 204 210 2-3 2-4 2-5 2-6 4~45 LIST OF FIGURES Real and imaginary parameters of the pion-nucleon scattering amplitude, isoscalar (isovector) shown with a solid (dashed) curve. Calculation of pionic atom shifts and widths using Set B, which was fit to these data. Elastic scattering with 50 MeV (left) and 162 MeV (right) n+ calculated with parameters from sets A, (impulse approximation), B and 3', shown with dashed solid and dash-dot curves, respectively. Real and imaginary parameters used to describe absorption of a pion on two nucleons, isoscalar (isovector) parts shown with a solid (dashed) curve. Pionic atom observables calculated with sets C (theory) and D, shown with dashed and solid curves, respectively. Elastic scattering calculated for SO and 162 MeV w+ using parameters from sets C, D and D', shown with dashed, solid and dash-dot curves, respectively. Elastic and inelastic scattering of 50 MeV n+ from 12C and its 4.44 MeV (2+) state, using optical parameters from Set D and an inelastic scattering t-matrix (plotted on the right) defined with parameters from sets A, B and 3', shown with dashed, solid and dash-dot curves, respectively. Same as Figure 4-1, except the t-matrix was calculated with parameters from sets C and D, shown with dashed and solid curves, respectively. Elastic and inelastic scattering of 162 MeV n+ from 12C and its 4.44 MeV (2+) state using optical parameters from Set D and an inelastic scattering t-matrix (plotted on the right) defined with parameters from sets A, B and 3', shown with dashed, solid and dash-dot curves, respectively. Same as Figure 4-3, except the t-matrix was calculated With parameters from sets C, D and D', shown with dashed, solid and dash-dot curves, respectively. Plots of lsz|2 for elastic scattering of so and 162 MeV r+ with parameters from sets A, B and 3', shown with dashed, solid and dash-dot curves, respectively. Elastic and inelastic scattering of 50 MeV 1r+ from 120 and its 4.44 MeV (2+) state using Set D to calculate the inelastic scattering t-matrix, while the optical used parameters from sets C (dashed curve) and D (solid curve). 15 24 27 34 39 43 78 80 83 86 89 92 4-10 4-11 4-12 4-13 4-14 4-15 4-16 4-17 4—18 4—19 Same as Figure 4-6, except the optical potential used parameters from sets A, B and B', shown with dashed, solid and dash-dot curves, respectively. Same as Figure 4-6, except the same parameters were used for both the Optical potential and the inelastic scattering t-matrix, either Set B (dashed curve) or Set D (solid curve). Elastic and inelastic scattering of 162 MeV n+ from 12C and its 4.44 MeV (2+) state using Set D to calculate the inelastic scattering t-matrix, while the Optical potential used parameters from sets C, D and D', shown with dashed, solid and dash-dot curves, respectively. Same as Figure 4-9, except the Optical potential used parameters from sets A, B and 3', shown with dashed, solid and dash-curves, respectively. Same as Figure 4-9, except the same parameters were used for both the Optical potential and the inelastic scattering t-matrix, either Set 3' (dashed curve) or Set D (solid curve). Elastic and inelastic scattering of 36 MeV n+ and n‘ from 120 (tOp row) and 2831 with parameters from Set C (dashed curve) and Set D (solid curve) as described in the text. Elastic and inelastic scattering of 50 MeV n+ from 12C, 2881, 40Ca and 208Pb with parameters from Set C (dashed curve) and Set D (solid curve) as described in the text. Same as Figure 4-13, except calculated for 50 MeV n’ scattering. Comparison of |S£|2 for elastic scattering Of 50 MeV n+ (tap) and n’ from 208Pb. Elastic and inelastic scattering of 67 MeV n+ and n’ from 12C using parameters from Set C (dashed curve) and Set D (solid curve) as described in the text. Elastic and inelastic scattering of 80 MeV “T from 12C, 40Ca, 90Zr and 208Pb using parameters from Set C (dashed curve) and Set D (solid curve) as described in the text. ‘ Same as Figure 4-17, except calculated for 80 MeV n“ scattering. Elastic and inelastic scattering of 116 MeV fi+ and n‘ from 40Ca (tOp row) and 208Pb using parameters from Set C (dashed curve) and Set D (solid curve) as described in the text. vi 94 96 99 101 103 108 111 113 116 118 121 123 128 Figpre 4-20 4-21 4-22 .5-m8 Elastic and inelastic scattering of 130 MeV'n+ and n“ from 283i using parameters from Set C (dashed curve) and Set D (solid curve) as described in the text. Elastic and inelastic scattering of 162 Mve+ and n" from 12C using parameters from Set C (dashed curve) and Set D (solid curve) as described in the text. Elastic and inelastic scattering of 180 MeV'n+ and n- from 2881 (tOp row) and 40Ca using Set C (dashed curve) and Set D (solid curve) as described in the text. Radial transition density (tOp left) and longitudinal form factor (tOp right) for collective (solid curve) and microsc0pic (dashed curve) models of the 40Ca (3.74 MeV) 3‘ state, the 50 MeV n+ inelastic scattering calculations at the bottom are described in the text. Inelastic scattering of n+ and-n- from the 3' state 40Ca at 116 MeV (top row) and 180 MeV, using the microscopic density with sets A and C (dashed and dash-dot curves, respectively) and the collective model with Set C (solid curve). Radial transition density (tOp left) and longitudinal form factor (tOp right) for collective (solid curve) and microscoEic (dashed curve) models of the 12C (4.44 MeV) 2 state, the 50 MeV n+ inelastic scattering calculations at the bottom are described in the text. Inelastic scattering of n+ and n‘ from the 12C 2+ state at 68 MeV (tOp row) and 162 MeV, using the microscopic density with sets A and C (dashed and dash-dot curves, respectively) and the collective model with Set C (solid curve). Longitudinal form factor (tOp) and 162 MeV n+ and n— inelastic scattering from the 2881 (9.70 MeV) 5‘ state with the two form factors described in the text. Transverse electric form factor (tOp) and 162 MeV n+ and "‘ inelastic scattering from the 208Pb (6.10 MeV) 12+ pure neutron state using Set A (dashed) and Set D (solid curve) parameters. The top row shows elastic and inelastic scattering of 180 MeV «+ and n‘ from 48Ca and its 3.83 MeV (2+) state with Set C (dashed) and Set D (solid curve), the bottom row compares 180 MeV n+ (solid curve) and n" (dashed curve) scattering from 40Ca (3') and 48Ca (2*) with Set D. + Trransverse magnetic form factor (tOp) and 162 MeV n and n‘ inelastic scattering from the 2881 (14.36 MeV) 6‘ state with the force and microscopic form factor described in the text. vii Page 131 134 136 143 146 148 151 154 158 161 165 Figure 5-9 5-10 Transverse magnetic form factor (tap) and 180 MeV n+ and n' inelastic scattering from the 12C (15.11 Mev) 1+ state, as described in the text. Transverse magnetic form factor (left) and 162 MeV w+ and n‘ inelastic scattering from the 16O (18.98, 17.79 and 19.80 MeV) 4' states, as described in the text. Angular distribution of single charge exchange with 50 MeV and 162 MeV n+ on 13C, using the Lane model (dashed curve) and single particle model (solid curve). Excitation function for n+ single charge exchange on 13C, using Lane (dashed) and single particle (solid curve) models. Dependence of 100 MeV n+ single charge exchange on target mass using the Lane model. Single charge exchange with 50 MeV n+ on 15N, calcu- lated with the Lane (dashed) and single particle (solid curve) models. viii 169 172 182 184 187 189 Table 1-1 2-1 2-3 2-4 4-1 4-2 4-3 4-4 LIST OF TABLES Definitions of Symbols Used in This Work Parameter Set A, Impulse Approximation Values for the Four-Parameter Optical Potential Parameter Sets B and B', Fitted Values for the Four-Parameter Optical Potential as Described in the Text Parameter Set C, Multiple Scattering Theory Values for the Second-Order Optical Potential Parameter Sets D and D', Fitted Values for Second-Order Optical Potential as Described in the Text Reaction Cross Sections in mb for 162 MeV w+ Elastic Scattering Parameter Set C Theory Values for Low Energy Scattering Parameter Set D Fitted Values for Low Energy Scattering Parameter Set C Theory Values for Resonance Region Scattering Parameter Set D fitted Values for Resonance Region Scattering RPA Vector and Transition Density for 40Ca and 12C Transition Densities for 2831 5‘ State SpinrOrbit Parameters from Rowe, Salomon and Landau Transition Density for 120 1+ State SpectroscOpic Z Coefficients for the 16O 4“ States Cross-Section Ratios (fl+/fi‘) for 16O 4‘ States Density Parameters ix Page 21 21 37 37 44 105 105 126 126 141 156 163 167 170 173 199 CHAPTER 1 INTRODUCTION This work arose from the need to deve10p a theory which would enhance the use Of pions as effective probes Of nuclear structure. Experimental facilities have improved greatly in recent years, leading to a need for a quantitative description of pion inelastic scattering. The energy resolution and beam intensity of early pion experiments limited studies to isolated strong states of easily manufactured targets. The advent of the large meson factories (LAMPF in Los Alamos, TRIUMF in Vancouver, SIN in Switzerland), which were designed to produce useful beams of pions from high currents of intermediate energy protons, led to the study of a wider range of reactions and the discovery of some previously unknown states. The theory Of pion-nucleus scattering fol- lowed a similar pattern, as SOphisticated theories became necessary to explain the more precise experimental results. The pion was predicted to exist in 1935 [Yuk 35], and after its discovery around 1947 it was soon applied to nuclear physics experiments. One Of the earliest experiments measured the angular distribution Of 52 MeV 1r+ and 1r" scattered from 12C [Byf 52] in a cloud chamber. The development Of scattering theory [Wat 53, Gel 53] found applications in photomeson production [Fra 53], nucleon-nuclear pion production [Kov 55] and the msslinger model [Kis 55] for pion scattering. The Kisslinger model was invented to explain the original data of Byfield 3.3 31., and later data [Sap 56, Bak 58] were found to require [Bak 58a] this form of the Potential . Another surge of activity followed the experiment of Binon E31. [Bin 70] which measured pion scattering from 120 to 280 MeV at CERN. Two calculations Of the inelastic transitions were soon published, one with collective [Edw 71] and the other with microscopic [Lee 71] models for the transition density. These assumed the Distorted Wave Born Approxi- mation (DWBA) [Sat 64, Aus 70] and used the impulse approximation (IA) [Ker 59] for the pion-nucleus interaction. This thesis continues the develOpment begun at that time, but with greater emphasis on the impact of recent improvements in the pion interaction on such calculations. Once accurate data began accumulating, particularly for pionic atoms and low energy scattering, the need for higher order corrections to the Kisslinger theory became clear. The most important of these was the so-called Lorentz-Lorenz Ericson-Ericson (LLEE) effect [Eri 66]. A number of Others, including phenomenological absorption terms, are outlined in the review article of Hufner [Huf 75]. Several more recent review articles [Bro 79, Tho 80, E13 80] are useful background to the work in this thesis. It should be emphasized that the point Of view here focuses on Many others are possible. A momentum a coordinate space potential. space potential is used by Lee [Lee 74, Lee 77]. The delta-hole model [311‘ 79, Ose 79, Gas 79a] is widely used in a microscopic description of P1011 scattering. Although neglected here, these other models are in'POI'télnt to the understanding and interpretation Of the Optical model results in terms of the fundamental pion-nucleon interaction. The experimental situation is characterized by the same diversity. Data exists for all pion reactions for some beam energy and target com- bi nation: pion angular distributions for specific final states via ;‘255v a a p .u elastic and inelastic scattering, single and double charge exchange reactions; inclusive measurements such as absorption, reaction, quasi- elastic and total cross sections; excitation functions and angular distributions for other quasi-elastic reactions involving nucleon knock- out which result in specific final states. As data accumulates for a systematic collection of targets and beam energies, an important role of the theory is to build a coherent picture out of the many pieces. The purpose Of this thesis is to examine the inelastic scattering data with the goal of understanding the prOperties of the effective pion-nucleus interaction. The starting point is a theoretical potential, develOped elsewhere [Str 79, Str 79a], which incorporates various cor- rections to the IA result for the pion-nucleon interaction. The elastic scattering data are used to identify deficiencies in this theoretical potential and fix the strength of the isoscalar interaction. Comparison wit}: the fit to a four-parameter potential allows the identification of general properties of this effective potential. These results are then used to calculate the inelastic cross sections, testing the interaction strength. This approach can then be extended to study isovector and spilr-flip transitions, which are not determined by the elastic scattering. This thesis divides naturally into two parts: the development Of the equations and forces to be tested is done in the next two chapters, While the applications to inelastic scattering are given in the follow- m3 t:hl‘ee chapters. C[l'lapter 2 reviews the multiple scattering formulation of the pion- nucleus Optical potential. Two forms Of the potential are used, one related to the simple four-parameter Kisslinger potential, the other con- ta1 ' fling all of the kinematic and other corrections used [Str 79a] to ¥ explain the elastic scattering. Each Of these is fit to the elastic scattering data to define a phenomenological effective parameter set to complement the theoretical values. Chapter 3 reviews the derivation of the DWBA and works out the specific equations needed for these calculations. Chapter 4 is the focal point of this work. Here the collective model is used to test the prOperties of the interaction and the effects of the various changes introduced to improve the fit to elastic scatter- ing. In addition, the effects of different choices for the distorting potential are examined independently of the interaction that induces the inelastic transition. Chapter 5 introduces the microscopic model of the transition density, and also examines the spin-orbit interaction for S=l transitions produced with pions. Chapter 6 introduces single charge exchange reactions, and gives some results as a means of examining the isovector part of the pion-nucleus interaction. These results are summarized in Chapter 7, where the general pToperties Of the effective interaction are used to give perspective to the discussion. There is much data, each with some systematic error, 3° it is only with a consistent approach that one can form these results int° a unified whole. The notation used here is listed in Table 1-1. Most of it is conveIltional, with the main exception being the ordering Of the argu- hence in the symbol for a Clebsch-Gordon coefficient. The angular In("Wuhan algebra follows Brink and Satchler [Bri 75]- xv. an. 7.x Table 1-1 Definitions of Symbols Used in This Work Kinematic Variables + k. + P. E primes subscript cm subscript 2cm subscript ACM no subscript u M “N 6 Others ¢ 0(r) 60(r) 8n Aguigigr MOmentum <1“ I-MIJMJ) xCabc, def, ghi) Ln 3-momentum and total energy of pion 3-momentum and total energy of struck particle indicate variables for final state pion-nucleon center of momentum system pion-two nucleon center of momentum system pion-nucleus center of momentum system ACM unless otherwise noted pion rest mass nucleon rest mass rest mass of nucleus ; AM reduced total energy in projectile-target center of momentum system plane wave total scattered wavefunction = O + x scattered wave, also used for the "distorted" wave a wave scattered by Optical potential only pion-nucleon scattering amplitude pion-nucleon t-matrix antisymmetrization operator number Of nucleons = Z+N isoscalar ground state density normalized to A nucleons (= on + op) isovector ground state density normalized to Z-N nucleons (=I pp - on) pion charge same as<'defined in Brink and Satchler a b c - d e f g h i =' \/ 2J+1 - 9-J symbol st.-. out nu...‘ "in”. .e. Q . . t. ‘c I ‘e. .O.. e 4“ C .0 41‘ ll.’ CHAPTER 2 THE PION-NUCLEUS INTERACTION The pion-nucleus interaction is a many body problem and there are a number of approaches to its solution, some of which are described in the previous chapter. Our approach is to reduce the problem to that of the interaction of a single particle with a potential which describes the average properties of the actual pion-nucleus interaction. This potential is derived using multiple scattering theory, which says that the pion-nucleus potential can be related to measured and calculated properties of the pion-nucleon interaction. The derivation used here follows that used by Stricker [Str 79a] and Brown [Bro 79], with an emphasis on those parts which are critical to the interpretation of inelastic scattering. The first section outlines the multiple scattering theory as applied here. The remaining sections break into two distinct parts. In SeCtions 2.2 and 2.3 the first-order potential is obtained from the Pionvnucleon scattering amplitude, and then compared with elastic scattering data. In Sections 2.4 and 2.5 the second-order terms are it"Reduced and then compared with the data. In each case a purely theOratical potential will be presented, and then a potential fitted tx) the data will be obtained. This will give us the flexibility to compare phenomenological potentials with theoretical ones at various stageS in the distorted waves analysis of inelastic scattering. Sec- t 1°“ 2.6 will sumarize these results. 2. 1 MULTIPLE SCATTERING THEORY The purpose Of this section is to develop a working basis for t}: e calculations and discussions which follow. The theory here is 6 ‘ s. It:- . vs. S . in. .‘ . 'ItJ . '51.» . “In. "hrs. ,.a . D..‘. ". \.‘a: based on the work Of Watson [Wat 53] and Kerman, McManus and Thaler [Ker 59], among others. The starting point for the description of a wave V scattered by a Hamiltonian H = Ko + V, where K0 is the kinetic energy and V the interaction with the scatterer, is the scattering amplitude fab = --;:§;§-<¢blv va> . (2-1) The scattering amplitude is just the projection of the scattered wave onto the plane wave o in the Outgoing channel. The scattering cross section is then 35 |fabl . (2 2) It is convenient to describe this process entirely in terms of oPerators. We write v = ¢ + x = s + cv T (2-3) Where X is the scattered part of ‘P and G = (E - Ko + is)"1 is the Greens function for outgoing scattered waves. If we define ‘i’ = NO we get an Operator equation 9 = 1 + GV 9 (2'4) toreplace the equation for the scattered waves [equation (2-3)]. The tranSition matrix is defined by T . v o (2-5) 8" that VT 8 To and the definition Of the scattering amplitude becomes 5 m f s .. ——_<¢ lava) = - T . (2-6) ab 2n h2 b 2n hz ab Since T = VQ, equation (2-4) gives us an integral equation for T, T = v + VGT (2—7) which is known as the Lippmann-Schwinger [Lip 50] equation, and is the starting point for the derivation. Before we continue, however, it will be necessary to examine the wave equation for the pion. Since the pion is so light, it must be treated relativistically although the massive nucleus can be treated non-relativistically throughout. Then the Hamiltonian for the system is 1/2 2 H=(k2+u2) +MN+—3—+v (2-8) ZMN Where k, p, P, MN are the momentum and mass of the pion and nucleus, respectively. The Schrodinger equation for this system, RV = BTW, was Shown by Goldberger and Watson [Col 64] to be equivalent to a Klein- Gordon-like equation 2 2 2 P N This can be simplified to (V2+k: - 28v+g—VZ)T= 0 (2'10) Where <3 :- wE/(w + E) is the reduced energy and kg- 002 - 112- If we separate out the electromagnetic potential and drop the smaller terms we are left with EIEI < 0 1r EM EM) ( ) Returning to the Lippmann-Schwinger equation, it will be con- venient to work with the equivalent equation 2w T = 25 vTr + 2; vflc 25 T (2-12) where G = (-k2 + k3, + is).1 with momenta defined in the n-nucleus system. The remainder of this section will show how this equation can be rearranged so that all the important (large) terms can be integrated over the target, thus reducing this to an equivalent one-body equation. Kerman, McManus and Thaler [Ker 59] showed that it is particularly convenient to use antisymmetrized intermediate states. We can do this by introducingfifl, an operator that projects onto totally antisymmetric target states, into equation (2-12) as is also done in [Mac 73]. This 18 possible since 5 commutes with G and V and since we will only take matrix elements of T between correctly symmetrized nuclear states. Then our starting equation is T = v + vc(42c’s)T ‘ (2-13) For convenience, the factor of AZ?» will be suppressed from here on except when it is important for a particular result. The first step is to define an auxillary potential matrix by U a v + VGQOU (2-14) Where Qo - l - P0 = l - |O><0' projects off of the ground state. It is the assumption of what follows that this subseries converges rapidly Since the matrix elements connecting the initial (ground) state to eReited states should be smaller than the diagonal matrix elements. 10 This allows us to rewrite equation (2-13) by using V = U(1 + GQOU)'1 so that U(1 + GQOU - GQOU)(1 + CQOU)~1(1 + CT) 1-3 ll U(l + GT) - ucqouu + GQOU)'1(1 + CT) U(1 + GT) - UGQOT U + UG(1 - Q0)T (2-15) Which gives us the integral equation T = U + UGPOT (2-16) for ‘I‘ in terms of U. Returning to equation (2—14), we now use that V =Zv, where v is the interaction with one nucleon, and the sum runs over all N target Particles. Because of the Operator A in the definition, all of these n“Cleans are equivalent and we can write U = Nv + Nv GQO(;325)U .4 (2-17) We now define an effective two-body operator T by T = v + vGQo‘r . (2-18) If We use v a 1-(1 + GQOT)‘1 in equation (2-17) we get NT(1 + GQOT)-1(l + GQOU) c: ll NT(1 + GQOU) - NT GQo v(1 + GT) NT(1 + GQOU) - NT GQoU/N N1 + (N - 1)T GQOU . (2-19) v ,- u ..- v ,‘ ‘v- 11 (2-20) If we define and U' = (N — 1) ’I' + (N - 1)TGQOU' we get the multiple scattering series for U. It remains to relate the effective Operator T to the free pion- The latter is defined by (2-21) nucleon t-matrix. — v + v 2;) t g cm TTN tnN where g = (-k2 + k3 + is)-1 with momenta defined in the TT-nucleon SYStem. Again, by writing v = tflN(1 + g tum-l and substituting into (2-22) ) equation (2-18) we obtain m = +t A2_-— ’ T th "N(GQO m g ZmC The Impulse using the same algebraic technique as in equation (2-15). Approximation (IA) assumes the higher order terms in equation (2-22) (2-23) Can be neglected so that a t . T TTN This approximation will be used throughout this work. With this assump- tion, the multiple scattering series for U can be written as TIN (2-24) 2wU = N 2w tnN + NCN - 1) 2m thN GQo i62¢» t 2 .. _ + N(N - 1) 2w tflN GQo £32m tflN 12 This series for U will be the basis of our analysis of the pion-nucleus interaction. This work will not focus on the specific effects of each of the terms in equation (2-24), as these are discussed in Stricker's thesis [Str 79a], but will restrict the discussion to two broad cases. In Section 2.2 the first order part of this, 25 U = N 25 cm , (2-25) Will. be examined. In Section 2.4 some of the higher order terms will be evaluated and summed to give the second order corrections to the int eraction. 2 -2 THE FIRST ORDER POTENTIAL The starting point for this analysis is the measured pion-nucleon Scattering amplitude. A convenient parametrization is = b + t- + fTTN (o b “~r)+(c0 c (2-26) Where t and T are the pion and twice the nucleon isospin Operators, .,_ .. .. k 13 the pion momentum in the center of mass (to be written kcm from her + e on), and O' is twice the nucleon spin Operator. Appendix A defines the relationship between the parameters b1, c1, 31 and the corresponding p:Lo'fl‘uucleon phase shifts. These phase shifts are computed from the parametrization of Rowe, Salomon and Landau, [Row 78] who fit an analytic f “he-tion to the phase shift data below 400 MeV. This approach has the advantage of producing smooth, well-behaved results even where the data at e particularly noisy, as is the case in the $11 channel below 100 MeV. 13 The values of these parameters are plotted in Figure 2-1. Notice that Rebo (solid curve) is nearly zero at low energy, due to cancel— lation between the 311 and S31 phase shifts, and becomes increasingly repulsive with increasing energy. Rebl (dashed curve), the isovector s~wave parameter, is roughly constant and repulsive. Rel;o (dash-dot curve) includes second order corrections and will be discussed in Sec- tion 2.4. Imbo increases slowly while Imbl is small at all energies. In contrast, the p-wave parameters co (solid curve) and c1 (dashed curve) demonstrate simple resonance behavior, reflecting the dominance 0f the P33 channel due to the low-lying A(1232) resonance. The spin- Orbit parameters also vary rapidly as they are also dominated by the A33 resonance. The pion-nucleon transition matrix is simply related to the scat- tering amplitude by f? 9 a ' 2" 7 1rN - A <0\2;) t Nl0> - A <0|(21r)3 6(1t+p-'-k'-p') 4.1: :5- fflN]0> (2‘29) (1) cm 14 Figure 2-1 Real and imaginary parameters of the pion-nucleon scattering amplitude, isoscalar (isovector) shown with a solid (dashed) curve. .?) If r11 _ .,H._ _ W». ,3, ._~__—_. 44 15 0.1 A 0.0 E a: 5 -0.1 -O.2 1.0 a,“ E 5:, <3” MSUX -8l—O93 Reol Imoginorg _ .. b0 ]— L c l l l l L J 1 300 0 100 l 200 E [MeV] Figure 2-1 ex 16 where IO> " ‘l'(p, p2. '°'. PA) and <0l= \P (p', p2, °", pA). Since f"N is independent of p this can be reduced to - = —- ' _ 20° UOpt 4" fflN (k. k ) 0(q) (2 30) 6 cm where q = k'-k and the factor A has been incorporated into o(q). If we assume a nucleus with N=Z we can conveniently drOp the isovector terms. Further, the spin-orbit terms vanish for a closed-shell spin-0 nucleus, and will be small otherwise [Lov 81]. Then 25 Uopt (k. k') = -41r (b0 0(q) + co OCQ) k°k') E or (2-31) (b0 O(r) - V°[co p(r)] V) are the appropriate forms for the Optical potential. Other forms, par- 2w Uopt(r) = -4n w ticularly the local Laplacian form, are described in Appendix B. Although this is a reasonable representation for the first order potential, it is not really complete because the potential will be used in the Dion-nucleus center of mass (ACM) while an has been defined in the Pion‘nucleon center of mass (cm). The transformation needed is not well defined in this case because the interaction is not written in terms 0f invariants. Relativistic potential theory [Lan 73] gives one Prescription for this, and this widely used [Tho so, Bro 79] method will 818° be adopted here without further argument. Its effects are dis- cussed in [Str 79a]. The prescription is to use - -411(21r)3 O(k'+p'-k-p) :9— y fflN(k(':m, Item) (2-32) (1) cm l7 Ecm wcm cm (”cm (”cm where ‘Y = E E' ' 3 —- w m m and fflN is evaluated at the pion-nucleon center of mass energy. The reader is reminded that unsubscripted.variables are in the ACM system in the remainder Of this thesis. Since - w - m ~ cm m 1+8 Y —— = —— —— = , (2-33) 5 (A) a; 1+E/A CID CID the only change is the addition of this factor that multiplies the entire potential. It is also necessary to express the momenta kcm: kém in the pion-nucleus center Of mass as well. This will affect both the p-wave + and the spin-orbit terms. The Lorentz transformation for kcm is + + + Y + + kcm=k+8 Y+lfi3'k--m (2-34) + _). where B =- (k + p)/(E + m), with a similar result for kém. To first order this simplifies to + -> -> + + . kE-Ew ; k-EE . _ kcm E+m 1+8 (2 3 5) A Particularly convenient way to do this "angle transformation" is to rewrite the equations in terms of + 1 + .y. + + + p'§‘ PW“? (2-36) + + + -) -)- + + Q - k+k' q - k'-k ' “’0 Since the equations corresponding '30 (2‘35) are + + + + '2' 9 26 . a - —- . 2" 7 qcm q and Qcm 1+6 1+8 P ( 3 ) a r— s.» _. —- ‘g. we. _ p,‘ "‘ \ a. .g‘: D n -." 5 18 Then the transformation gives + ‘* 1 2 2 o ' a —- .. kcm kcm 4 Qcm qcm (2-38) 2 2 + -> 82 3 ii? - %—- " 8 P'Q + 2 P2 (1+8) (1+8) (1+8) so that 2 + ->- + -> k 'k' = 1 2k-k'- 8 25;— cm cm (1+8) (1+8) (2-39) —>- + 2 _ 8 2 P'Q + 8 2 P2 (1+8) (1+8) The integration over the ground state momenta, called Fermi averaging, causes [Bro 79] the third term to vanish and the fourth term becomes pro- portional to the kinetic energy density in the Thomas-Fermi approximation, 2/3 K(r) =§<§ “2) 95/3 . (2-40) This term, which acts like an attractive s-wave term in the potential, will be ignored, as in [Str 80], by incorporating it in the second order corrections added in Section 2.4. In the fixed scatterer approximation it 13 iSnored altogether. The transformation of the spin-orbit term involves + 7" + + -> ->- g + -> kxk' -i =l_._1 -__p 2-41 cm cm 2 anx qcm 2 1+8 Q x q 1+8 x q ( ) 80 that + 11" ———l f: 12" + —E + h (2 42) a X 0 - kcmx cm 1+8 x +€ p p 19 The second term will be drOpped. This term can contribute to elastic scattering from nuclei with spin-unsaturated subshells [Lov 81], such as 48Ca. In addition, it can affect normal parity inelastic transi- tions, but will not be a significant part of the interaction for the states considered in this work. Notice that the spin-orbit force comes in with one less factor of (1+8) in the kinematics. Since this part of the potential only contributes to S=l inelastic transitions, we will not return to it until much later. If we collect together what we already have and include the iso- vector terms, we get the first order Optical potential that will be used here. It is 25 Uopt(r) - -41r{p1 [b0 p(r) + eTr bl 60(r)] pl +-12-(1 - p11) V2 [coo(r) + e:Tr O1 6900]} (2-43) V- [CC p(r) + 81r cl Op(r)] V where P1 =' (1+8)/(1+8/A), E" =- pion charge, Op(r) = OP(r) - O (r) and n 0. PP. on are normalized to A, Z, N, respectively. The densities are usually assumed to have the same radial dependence so that \ cTr 60(1’) = c1T 3%"- pm . (2-44) f 2.3 CALCULATIONS WITH FIRST ORDER OPTICAL POTENTIAL { The Properties of this potential will be illustrated with a series Of calculations at a representative set of energies -- zero, 50 MeV and 162 MeV -- for a range of nuclei. The zero energy calculations are for the shifts and widths of levels in pionic atoms. The calculations use a modified version of MATOM [Seki] and are compared to the available 20 data. The other calculations are for elastic scattering from 160, 400a (at 50 MeV only), 208Pb. These results are obtained using a modified version of the program PIRK [Eis 74]. The density parameters for these calculations are taken from electron scattering. Tables are given in Appendix C. Three sets of calculations will be described. The first, desig- nated as Parameter Set A throughout this work, uses the phase shift values for the parameters b0, b1, co, c1 in the Optical potential defined in equation (2-43). These are listed in Table 2-1. This illustrates the effect of the lowest order estimate to the Optical potential. The second, designated as Parameter Set B, results when fitted values for Rebo and Imbo, (at 50 MeV only) Reco and Imco are used. The fitting procedure was to minimize the average x2/point for all the data available at the given energy. These parameters are listed in Table 2-2. An auxillary Set, 8', as defined at 50 MeV, varies only 3 independent parameters since the ratio of Imbo/Imco was held fixed at the value determined by pionic atoms. These sets illustrate the effective potential required by the data at these energies which will be valuable when we look at the effects of the second order corrections in the next part of this chapter. A different Set B' is defined at higher energies. In this case the fit varies Reco and Imco as before, ex“Pi: the V2 term in the Optical potential is omitted. This set allows a comparison to the potential used by Holtkamp and Cottingame [Cot 80]. The first illustration of the effects of these potentials will be taken from Picnic atoms. Measurements of the strong interaction shift of level eneI-‘gies for Is [Tau 74] and 2p [Bat 78] levels provide infor- ma . o o tlon on the real parts of the s- and p-wave potentials, reSpectlvely. 21 Table 2-1 Parameter Set A, Impulse Approximation Values for the Four-Parameter Optical Potential Tr-Atom 50 MeV 162 MeV b() -0.006 -0.030 + 0.019 1 -0.079 + 0.041 1 b1 -0.133 -0.131 - 0.005 i -0.125 + 0.005 1 co 0.65 0.75 + 0.090 1 0.36 + 0.96 1 Cl 0.47 0.45 + 0.044 i 0.21 + 0.48 i Table 2-2 Parameter Sets B and B', Fitted Values for the Four-Parameter Optical Potential as Described in the Text Tr-Atom 50 MeV (B') 50 MeV 162 MeV 162 MeV (B') +0.0148 i +0.017 i +0.0064 1 +0.04]. 1'. +0.04l 1 C0 0.513 0.550 0.557 0.58 0.72 +0.0343 i +0.038 i +0.091 1 +0.66 1 +0.68 1 The correSponding widths of these levels provide information on the imaginary Parts of the s- and p-wave potentials. Since the potential under consideration has only four parameters (the isovector parameters are kept fixed), the effect of each one is easily identified. Specifi- tally, the negative value of Rebo produces the repulsive shift of the S-wave levels while the positive value Of Reco produces the attractive shift of the Prwave levels. The level widths result from absorption Whi ch is modeled by the imaginary parts of b0 and co. This makes the ‘7. a .as . r 22 fitting procedure straightforward -- the values of b0 and co are varied to minimize the x2 for the s- and p-wave shifts and widths separately -- although there is an interdependence between some of these parameters that requires careful checking of the fit. The results of this fit are given as Set B; the calculated shifts and widths are shown in Figure 2-2. It is not possible to calculate and plot similar results for the values of Parameter Set A as they do not produce a solution for a bound state in MATOM. However, the tabu- lated values for Set A are sufficient for a comparison. A reference to Tables 2-1 and 2-2 makes it immediately clear that the IA value for Rebo is much too small to explain the s-wave repulsion required by the data. In contrast, the IA value for Reco is too large to explain the p-wave attraction indicated by the data. Since the IA potential is ‘purely real at zero energy, Set A cannot explain the absorption of the Ilion in the nucleus. These phenomena -- increased s-wave repulsion, decreased p-wave attraction, true absorption of the pion -- are critical ‘33 ‘the understanding of low energy pion scattering. Much of the study (if. the second-order potential is directed towards explaining these pl‘Gperties. The other illustrations of the application of these potentials are taken from elastic scattering. These measurements are no less sensitive t" ':11e different pieces of the potential, although the effects cannot be Qlearly separated as is the case for pionic atoms. The low energy (50 I“14eV) scattering is dominated by the interference between the s- and p‘w‘ave potentials, which shows up as a minimum near 60° in the elastic ‘Sc: a‘tt-ering angular distribution. Furthermore, absorption dominates the 23 Figure 2-2 Calculation of pionic atom shifts and widths using Set B, which was fit to these data. 24 MSUX—8L094 11 30. 10 3. l. p-pph. » b-p-ppp O o 1 _>a: mud- 100. 30.” 3." 1 dqdqa‘dd 4 —»_—._ p p d - —«q44-d u q daqq‘ « u 20 30. —...-ppr - C 1. 3 :2. at .03 29 20 20. -1 Figure 2—2 V\‘ 25 reaction cross section and has important effects on the elastic scat- tering process. Higher energy (162 MeV) scattering is dominated by the p-wave (A33) resonance, so co is the important parameter. Since the nucleus is quite "black" at these energies, diffraction effects dominate the angular distribution. Sample calculations for sets A, B and 3', described below, are shown in Figure 2-3. The calculations using IA values (Set A) are shown with dashed curves. At low energy these results are clearly wrong. The IA values of Rebo and Reco produce an interference minimum near 75° and the mag- nitudes are incorrect. 0n the other hand, the high energy results are much closer to reproducing the data. This occurs because the potential has sufficient absorption to produce the diffractive scattering that is observed. Several authors [Thi 76, Joh 78] have remarked on the fact that a sharp cutoff model will reproduce the angular distribution at small angles. The calculations using fitted potentials require some discussion. lime fits are global in the sense that a single set of optical parameters “"313 determined by simultaneously fitting all nuclei for which data were aVailable. The average x2/point was minimized for the entire data set. At 50 MeV there were seven nuclei used: 12C, 160, 283i, 56Fe from Dyt- man 3.911.- [Dyt 79] and 120, 160, 400a, 9021-, 208% from Freedom 3511, [Pre 81]. At 162-163 MeV there were six nuclei available: 12C from Piffaretti _e_t_£l_. [Pif 77], 2831, 58m, 208Pb from Olmer e_t_a_l. [Olm 80] 6111:: 160, l'OCa from Ingram gt 2£° [Ing 78]. The 50 MeV fits were obtained for a four-parameter fit (Set B) Eltl‘i' ‘a.three—parameter fit (Set 3') where the ratio of Imbo/Imco was f1 GajL‘i constant at the pionic atom value. These are shown with the solid 26 >Hm>wuomamou .mm>u:o uovlsmmv mam mwaom .vonmmv suds asonm ..m use m .Acofiumefixouamm omaemaav < mumm aoum muouoamuma Lugs woumasoamo += Auemsuv >mz Nefi new Auumav >mz on as“: mcsumuumom ofiummam mu~ seamen MSUX-Bl-O95 27 p h- p. Wu"! r 111"“? r Intrttr I Inn”! I [nun I T [mutt 1 [Hunt I pvnnr I lnnyrv I 14 30 120 150 GCJnldeQ] 80 O m " -4 L . O O F‘ O o F. o e O O "9 o F. O I d H o [JS/qw] ap/op Trf t V [VITrr‘ r U 15", I r I I I'VIII I r I 'YTT' I I T T [11"] Y I I b 120 150 90 60Jn1d99] l 111111 1 1 '11111 1 L L l1x\1L1 L 1 I1111L1 1 1 IAQ‘JLL1 1 111111 1 L \\ O O O O O O 0-0 on 0'. (49/qu asp/op Figure 2—3 28 and dash-dot curves, respectively. Set B provides a slightly better fit but the difference is not very significant except at small angles, where the s-wave interferes with the coulomb amplitude. Notice that the improved agreement for these fitted sets comes about due to an in- crease in the s-wave repulsion and a decrease in the p-wave attraction, just as for pionic atoms. Notice also that the distribution of absorp- tion between s- and p-wave is quite different for Set B compared to Set 3'. Dytman [Dyt 77] obtained similar results for an unconstrained four-parameter fit, with all of the absorption going into Imco due to the lack of the v2 term [Yoo 81]. The difficulty with this is that it { lacks continuity with the pionic atom results. Set 3' shows that a good fit can be obtained without substantially altering the energy dependence of the parameters. Since the model to be develOped in the next section will use parameters that connect smoothly to pionic atoms, Set 3' will be useful for comparison. The 162 MeV fits were more difficult to obtain. Part of this Seems related to normalization differences between the data taken at different laboratories by different groups. The solution was to examine ‘i ifew'nuclei in detail; adjustment of the normalization of the 160 data 133’ 1102 produced consistent results. Others [Holt] seeking global fits have had to do similar things. The results are always presented without renormalization so the trends will be clear and not obscured by any E3QaJLing. For consistency, the data were only fit out to the second nmJLIIJLmum, since the data sets cover quite different angular ranges. The IEjL‘: with Set B (solid curve) is not substantially different from the r E283lalts with Set A (dashed curve); the location of the minima are not 29 correctly reproduced. The fit with Set 3' (dash-dot curve) is much better although the parameters are nearly the same as those in Set B. The increase in Reco and decrease in ImcO is consistent with the energy shift prOposed by [Cot 80]. This method is crude, but the results in Figure 2-3 serve to indicate the sensitivity of the calculation to variations of the parameters on this scale. We can summarize the low energy results by observing that the s-wave repulsion (b0) must be increased and the p-wave attraction (co) decreased. The absorption at 50 MeV is about 60% of the pionic atom value. The agreement with the data at higher energies is much better, with the uncorrected first order potential doing a reasonable job. Dropping the V2 term improves the fit at low angles by making the nucleus look smaller; large angle cross section measurements may be a means to test whether this is a reasonable form of the potential. 2.4 HIGHER ORDER CORRECTIONS TO THE PION POTENTIAL This section will discuss the contributions to the pion nucleus linteraction from second and higher order terms in the multiple scatter- 3ing series [equation (2-12)]. There are essentially three changes that o<=<:ur. First, there is a second order correction to bo that increases tlme s-wave repulsion, working against the kinetic energy density term. Tl35.8 is important since RebO is so small at low energies. Second, it is possible to sum the series for the p-wave part of the potential, an effect first described by Ericson and Ericson. [Fri 66]. Known as the I‘<>1-entz-Lorenz Ericson—Ericson (LLEE) effect because of the analogy to t:l'1<=e‘Lorentz-l.orenz effect in dielectrics, it reduces the p-wave attrac- . - t:zl-(Dn. Third, the true absorption of a pion, which must occur on two or ng 4.. 1‘~ .IA 1 I a I ‘I! u‘l Q. ~ .' 5.5- I..., -—-.» §~A I. (I) I . 30 more nucleons to conserve energy and momentum, will be included. The absorption of pions is a dominant part of the cross section at low ener- gies and must be included in any discussion of the optical potential. The first correction is to add the second order contribution to bo. This is very important since the first order value for bo is nearly zero due to the cancellation between the 511 and S31 phase shifts. The second order part of the Optical potential is 4" p12.é:l.<}02 + 2bl€> I 0(r) (2-48) so that we can define - 2 2 = b - b 2-4 b0 0 pl (0 + 2131) I ( 9) where I involves the expectation value of the two-nucleon correlation function. The result at zero momentum is I = , (2-50) assuming the Fermi Gas Model for the correlation function, and decreases PaPidly as energy increases [Str 79a]. This parameter is plotted as the dash-dot curve in Figure 2-1, where its importance at low energies can be clearly seen. There are new results [McM 81] that suggest a further enhancement of So due to medium effects involving p-wave rescattering. There is a second order correction, exactly analogous to be, that <=c’“nretical parameters correct. 31 The most important second order correction is the Lorentz-Lorenz Ericson-Ericson (LLEE) effect, which is also the least well understood. First derived by Ericson [Fri 66], it is a result of summing the p-wave terms to all orders in the multiple scattering series. Assuming hard core repulsion between nucleons, the p-wave potential is + -1 4w A-l -1 m + v - --——-——— v 4" pl Cop 2|: 3 co pl 0] ° m (2-51) -1 + 4n pl c p + 2V0 0 V .41 A-l -l 3 A pl Co0 ‘The original derivation, in analogy to the electrostatic Lorentz- Lorenz effect, assumed the only effect was due to the "polarization" of the medium by the strong p-wave interaction through the A33 reso- nance. Subsequent calculations, particularly those by Brown [Bay 75], Weise [Ose 79], Eisenberg [Eis 73] and their collaborators, have in- cluded effects due to N, D, m intermediate states and finite range effects. These modify the form of the LLEE effect by introducing a parameter A in the formula -1 4" p1 coo(r) 1 (2-52) 1 +‘51 1 p - 3 1 cop(r) so that the original result corresponds to A = l. The consensus of recent calculations [Bay 75, Thi 76a] is that 1 < A < 2, with values around 1.6 to 1.8 being preferred [Ose 79] at low energy. The value of 1.6 will be used here as it falls in a range preferred by low energy data. These larger values serve to substantially decrease the p-wave 32 strength at low energies, a decrease that is important if agreement with the data is to be obtained. The value at higher energies is very poorly known, the choice of A = 1 reflects that the value of A should decline as energy increases and correlations become less important. Last but not least, the effect of true absorption must be included. The absorption of a pion by two nucleons is the dominant part of the reaction cross section at low energies. Early pionic atom analyses assumed that B 02(r) - C 3. 02(r) 3 (2-53) 0 0 would be a convenient parametrization, with the p2 terms reflecting the fact that a two-nucleon density is required. It seems reasonable to cast the absorption into this form, but others [Ose 79] strongly suggest that this is only true below 50 MeV. Since approximate calculations [Cha 79] of these parameters, based on Fermi gas wave functions for the struck nucleons, exist over the full range of energies to be studied, we will adept this form. Specifically, we take ->- -> B + C k ° k' (2-54) 0 o 2cm 2cm to represent the absorption, where k2cm is defined in the two-nucleon pion center of mass. The transformation from the pion-nucleus system to this system is the same as before except that (l+€) becomes (1+8/2). The parameters calculated by Chai and Riska [Cha 79] are given in Figure 2-4. The short dashed lines at low energy indicate the values determined by pionic atoms, as will be described below. Although there are good arguments both against and in favor of these theoretical 33 Figure 2-4 Real and imaginary parameters used to describe absorption of a pion on two nucleons, isoscalar (isovector) parts shown with a solid (dashed) curve. 34 MSUX-8b096 BO [mei] -0.8 C 3- /‘ i H : / \Imognorg m _ E 2: / \\ a: - / o : / \ U 1.2"- / \ ;/ 0.. 0 100 200 300 E [MeV] Figure 2-4 35 values, an empirical argument for them is that they have the behaviour that one would expect for absorption dominated by the A33 resonance. When the absorption is included explicitly, it becomes necessary to identify Imbo and Imco with the quasi-elastic cross section. When this is done it is necessary to reduce their phase shift value by a factor Q which accounts for Pauli blocking of inelastic scattering of the nucleon. Following [Str 79], this is evaluated from Goldberger's formula [G01 48] using Landau and McMillan's approach [Lan 73]. This factor is 0.31 at 50 MeV, 0.54 at 100 MeV and 0.72 at 162 MeV. Results for Oq.e. appear consistent [Str 79a] with the limited data that is available. When these new terms are combined with the first order terms already included in UO in equation (2-43), we get pt _ 2 .. .. 5 2w Uopt 4w {pl[bo o(r) + 8,, bl C(r)] + 132 BO 0 (r) + C(r) 'V° 417 1 +‘5— A C(r) + V +_% (1_p1-1) V2 [C0 p(r) + a" el 60(r)] l -1 2 2 +3 (hp2 1 V [Co 0 (1')]: (2‘55) where -1 -l 2 C(r) p1 [co DC!) + ETr Cl 6p(r)] + P2 Co 0 (r) 9 where it has been assumed that the p-wave absorption terms are modified by the LLEE effect. The inclusion of Co p2 in the LLEE effect compli- cates the analysis, since a change in Co affects the p-wave strength of co, and vice-versa. In addition, some authors [Huf 75] believe it 36 should be kept separate. From the point of view of this analysis, the difference is a moot point since it only affects the final magnitude Of parameters needed to produce the same scattering, and these parameters are not particularly well known. The tradeoffs that are required be- tween these two alternatives have been given elsewhere [Str 79]. 2.5 CALCULATIONS WITH FULL OPTICAL POTENTIAL The properties Of the full optical potential will be illustrated with the same series Of calculations used in Section 2.3. However, the increased complexity of the potential increases the number of options we have for choosing Optical parameter sets. One way of dealing with this complexity is to relate all Of these potentials tO an equivalent four-parameter potential, so that a common set of four effective strengths can be used to relate all of these potentials to each other. This will be done in the next section when these results are summarized. Two sets of calculations will be focused on at low energy, reflect- ing varying degrees Of adjustment in the parameters. The first, Set C, uses the phase shift values for So: b1, co, c1 and the Chai and Riska values for Bo and Co. We fix A.= 1.6 as listed in Table 2-3. This set is analogous to Set A and illustrates the effect Of a purely theoretical model for the Optical potential. The second, Set D, results when Rebo and Race, and the amount of absorption (with ImBo/ImCo held constant) are adjusted to fit the data in the same way as described earlier for Set 8'. Three sets are used at 162 MeV. Set C is defined using the Izheoretical values for b0, co, Bo, Co as described above, with A21. ESet D is the same except that Reco and ImCo are fit to the data, in Einalogy to Set B. Set D' is also a fit, except the V2 terms are omitted in analogy to Set 8'. The results are given in Table 2-4. 37 Table 2-3 Parameter Set C, Multiple Scattering Theory Values for the Second-Order Optical Potential fl-Atom 50 MeV 162 MeV -0.033 -0.045 + 0.006 1 -0.083 + 0.029 1 -0.133 -0.131 - 0.002 1 -0.125 + 0.003 0.65 0.75 + 0.028 1 0.37 + 0.67 1 0.47 0.45 + 0.013 1 0.21 + 0.33 1 1.6 1.6 1.0 0.007 -0.02 + 0.14 1 -0.15 + 0.28 +0.08 1 0.29 0.36 + 0.59 1 1.29 + 2.95 1 +0.34 1 Table 2-4 Parameter Sets D and D', Fitted Values for Second-Order Optical Potential Described in the Text W-Atom 0.70 1.6 0.007 +0.19 1 0.29 +1.06 1 50 MeV +0.006 i 0.75 +0.028 i 1.6 162 MeV +0.029 i 0.45 +0.67 1 162 MeV (D') +0.029 i 38 These sets will be compared in the same way as in Section 2.3, starting with the pionic atom calculations shown in Figure 2-5. The dashed curve shows Set C, which does reasonably well but fails to get the details correct. The parameters of Set D were fit as described earlier, except that ImBo and ImCo were varied instead of Imbo and Imco. The results with Set D, shown with the solid curve, are quite similar to the results with Set B in Figure 2-2. This illustrates that the fit results are not particularly sensitive to the form of the poten- tial. What is promising is that the values of the purely theoretical potential Of Set C are now quite close to those required to fit the data. Indeed, some recent results [McM 81] suggest that p-wave medium corrections to So will increase its value to that required by the data. The absorption parameters are low, but the theoretical situation is far from clear. Recent work by Saraffian [R13 80] indicates that medium corrections to the calculation of B0 will bring agreement with the fit value in Set D. Calculations by Weise and coworkers [Ose 79] have produced values of ImCo that are much larger than those used in Set C. One set gave the value of ImCo a 0.68 fm6, which is much closer to the number determined here. Their value of ReCo = 0.97 fm6 would drastically change the result for co, reducing it to 0.60 fm3. However, the value of A significantly affects these predictions since all of the p-wave parameters are interrelated by the LLEE effect. It is large ambiguities like this that force us to fix on a single set and adjust it to the data, with the understanding that future theoretical work may clarify the exact values for different parts of the potential but will probably not alter the overall strength of the real and imaginary parts of the potential. 39 Figure 2-5 Pionic atom observables calculated with sets C (theory) and D, shown with dashed and solid curves, respectively. 40 MSUX-Sl-O97 mum seamen 100. . ‘4-4 q a J a _.: 1e qqdqdda 4 q dAaqaqd a a 1...... r 12 3 .10 1 r 118 2 19 f. 19 2 18 Z I o 12 17 F 16 16 1 F 15 112-1 bpp_b p. » _.L.pr. .pp~.p- . - LL:~[. - O l o O I O m m 1 m w 3 1 a 1 2,... £4. 3.... do 41 The conclusions about low energy elastic scattering are similar. As can be seen in Figure 2-6, the results with Set C (the dashed curve) are a big improvement over those with Set A in Figure 2-3. This is primarily due to the increase in s-wave repulsion due to the use of Po: and the decrease in the p-wave attraction due to the LLEE effect. The solid curves show the result of a fit, producing Set D. This is a three-parameter fit, with Rebo, Reco and the absorption (ImBo/ImCo fixed) adjusted to fit the data in the same way as was done for Set 8'. A comparison of sets C and D indicates that the change in Po required is almost the same as for pionic atoms, -0.015 fm. The p-wave param- eters agree very well, but this is mostly due to the deliberate choice of A - 1.6. It is particularly interesting to note that the absorption re- quired is about 622 of the pionic atom values for Set D. The implica- tion is that the absorption parameters decrease as energy increases, a result contrary to the predictions of all theories. However, calcu- lations of absorption cross sections agree reasonably well with some recent absorption measurements [Car 81] if the absorption data [Nak 80] are systematically renormalized within the stated errors. Whether this correctly reflects the physical situation remains to be seen, as it will require additional experimental measurements. The situation at higher energy is less clear. The results with Set C are shown with a dashed curve in Figure 2-6. The change from Set A is not very great; although differences at backward angles are significant, there is little data in this region. Set D is fit in the same way as Set B, with the same renormalizations of the data, and the 42 .>Hm>quooemmu .mo>ueo uouunmmp use cwaow .vonmmv nufiz csonm ..o van a .0 meow Bouw mumuoemume wean: += >0: NcH man on MOM moumaaoamo weauOuumom OHummHm elm munwfim MSUX-Bl-O98 43 - 1 " 1 O .. "ID F‘ P ct O l.- "‘N L .-- b 4 “8 I fr 141 L 8 h T b _ 1., . -1 I- -1 11111 1 11111111 1 11111111 1 1111111AM 11111111 1 I111111 1 1 11111111 1 11111111 1 [111111LL 1111111 1 1 \x p. v-l H O O H O O "‘ O O o O "9 O H O o u-o H O IITrrI r I [IIIII I r I [TYTIII fir [IIIIrT—I I 1111111 I I [IIIII I I r .4 .1 O “W F. .1 O "N d q *8 1 do (0 .1 O “m 1 .1 1111 1 L 1 1111111 1 4L 11x91111 1 1 I111L11 L 1 15‘111 1 1 1111111 1 1 \\ \x 8 Q O O O Q N G c-v O c-O o-c o c-o 0-0 0-1 [JS/QUJ] 69/09 9°.m.[deg) ecxnldegl Figure 2—6 44 results are comparable. Removal of the V2 term improves the fit, as Set D' (dash-dot curve) shows. The shift of the minimum is about the same as observed for Set 8'. One reason that the calculations are similar is that the reactive content of the potentials is similar. Table 2-5 gives the reaction cross sections for these parameter sets. Although it is not always possible to decompose the reaction cross section into absorption and quasi-elastic cross sections, it is clear that the model used here produces reasonable values for the imaginary part of the Optical potential. A short summary of these results is that the increased s-wave repulsion (be) and decreased p-wave attraction (LLEE effect) obtained from the second order corrections do much to improve agreement with the low energy data. Further, it is observed that absorption is important, contributing markedly to back-angle cross sections, and that it is possible to define absorption parameters which are consistent with the Table 2-5 Reaction Cross Sections in mb for 162 MeV 1+ Elastic Scattering .139 4008 2089b Set A 517 914 1955 Set B 510 899 1966 Set 3' 510 901 1968 Set c 503 883 1930 Set 0 509 893 1957 Set D' 507 894 1962 45 trend in Cabs and Gelastic from zero to 50 MeV. The higher energy data are reasonably reproduced by all sets, although drOpping the V2 terms improves the fit. Both fitted sets have larger values of Reco and smaller values for ImCo, as was observed for the sets B and B'. 2.6 SUMMARY OF ELASTIC SCATTERING RESULTS The most straightforward way to compare these potentials is to formulate an "equivalent" potential so that changes in a few parameters can be related to the changes in the calculations. A convenient form is the simple Kisslinger form 2;) U = '47TE) 0(r) - ce E7).- p(r) ‘5] (2-56) eff ff so that terms of higher order in p are incorporated into s-wave and p-wave strengths. We use a simple ansatz to define S + beff ‘ p1 0 pz 0 eff and (2—57) -1 p1 co + p2 Co peff eff ' 41 -1 -1 1 + 3 [91 co + p2 Co peff] peff C where isovector terms are suppressed. This prescription does not pro- duce a four-parameter potential that will generate exactly the same result as the full interaction; the density dependence cannot be re- placed by a simple constant except within a small region of parameter space. Despite this restriction, the use of peff 3 0.7 no - 0.12 fm."3 provides a basis for a qualitative comparison [Str 80] of quite dif- ferent potentials. 46 This form is particularly useful for interpreting the tradeoffs that occur between parameters. For example, the pionic atom data deter- mined ImBO = 0.19 for Set D, hence ImBo Deff p2/pl = 0.021 which is only 30% greater than the fitted value for ImbO in Set B. In a similar vein it was observed above that increasing ReCO from 0.29 to 0.97 had to be compensated by a decrease of Reco from 0.70 to 0.61. Since the change in ReCo translates into (0.68) oeff pz/pl = 0.076, we see that the simple model explains 80% of this change. The LLEE term is much more complicated, but implicit differentiation can be used to relate changes in A to changes in the other parameters. For example, we can get d1 4n co Co W a 3— 0eff pl[F]-: +— peff] (2-58) which is 3.7 for Set 0 at 50 MeV. This compares favorably with the value of (1.4-1.6)/(0.70-0.75) = 4.0 deduced from comparing this set to the best fit set with A = 1.4 [Car 81]. These results serve to indicate how this equivalent potential provides a basis for a qualitative inter- pretation of the full potential in terms of a more easily understood four-parameter potential. In summary, the simple four-parameter potential with IA fails miserably at low energy. The s-wave repulsion (Rebeff) must be in- creased while the p-wave attraction (Receff) is reduced. Also, there must be a large amount of absorption added, although the distribution between s- and p-wave is not clear. The results with IA values in the full potential outlined in Section 2.4 are much improved. This occurs since the second order corrections all go in the direction required by 47 the data. The use of be increases the repulsion in Rebeff, while the LLEE effect substantially reduces Receff. Fits indicate that only small adjustments are required to remove the remaining discrepancies. The calculated absorption is still too small for n-Atoms, although reasonable at 50 MeV. It remains to be seen if the 60% decrease in the absorption parameters from zero to 50 MeV can be explained theoretically, it is clearly required by the data. The higher energy results were that the simple four-parameter potential is reasonable, but is improved by using p-wave parameters that correspond to a shift of the resonance by 20-30 MeV. The full potential gives better results to begin with, but is also improved by a similar adjustment in the p-wave parameters. Omitting the V2 kine- matic term makes the nucleus appear smaller, improving the fit to the low angle data, but decreasing the cross section at larger angles. If one is willing to adjust the size of the nuclear density [01m 80], a similar improvement in the fit would also result. More complete studies will be needed to sort out these differences. In summary, we have obtained four potentials that reflect varying degrees of theoretical sophistication and phenomenological input. Each is suitable for the exact calculation of the distorted waves needed for the inelastic scattering calculations, allowing investigation of the sensitivity of inelastic scattering to the distorted waves used. Fur- ther, they provide reference values for the pion-nucleon interaction that will generate the inelastic transition, allowing a test of whether inelastic scattering is as sensitive to second-order corrections in the interaction as elastic scattering is. CHAPTER 3 INELASTIC SCATTERING FORMALISM The analysis of inelastic pion-nucleus scattering usually makes use of the Distorted Wave Born Approximation (DWBA), based on the assumption that this is a direct reaction process involving the exci- tation of a single nuclear state without the involvement of other reac- tion channels. There are transitions for which this assumption is not valid; they should be analyzed in the Coupled Channels Born Approxima- tion (CCBA) which we are not investigating here. This introduces an ambiguity into the analysis of some states which will be pointed out when apprOpriate. This chapter will detail the theory and equations necessary for the analysis of inelastic pion scattering. First, the formal theory of the DWBA will be outlined. The expressions for the transition matrix are worked out, relating the cross section to matrix elements of the pion- nucleon interaction. These matrix elements are then worked out, separ- ating terms involving the pion-nucleus interaction and nuclear structure, which include the "physics" of the process being studied, from angular momentum factors that reflect the rotational properties of space. The formulae of angular momentum algebra follow the conventions of Brink and Satchler [Bri 75] with the exception of the notation for the Clebsch- Gordon coefficient, which is given in Table 1—1. Finally, the details of the form factor are presented for the specific models considered here. 48 49 3.1 OVERVIEW OF SCATTERING THEORY AND THE DWBA The DWBA has its origins in the Gell-Mann-Goldberger [Gel 53] relation for scattering from two potentials. The details of the formal theory of scattering used in deriving this result will not be given here, as they can be found in many places [Col 64, Aus 70, Jac 75, Sat 64, Sch 68]. However, a summary is presented here for completeness. In Chapter 2 it was shown how the cross section was related to the transi- tion matrix T, which satisfies the Lippman-Schwinger equation 1 = v + v01 . (3—1) The expansion of this gives T = V + VGV + VGVGV + °'° (3-2) which is recognized as the Born series for T. If we keep only the first term of this series, which corresponds to taking the first term of the series for Q - l + GT, we obtain the Plane Wave Born Approxima- tion (PWBA). This gets its name from the fact that r = <¢ajvj¢b> . (3-3) Although simple, this approach fails when V is strong, so that the series in equation (3-2) does not converge quickly, if at all. The Gell-Mann-Goldberger relation provides a way around this prob- lem when the potential can be broken into two parts, V - V0 + V1, where Vo << V1 and is known so that scattering by Vo can be calculated exactly. 50 We define T = V + V GT 0 o o o and (3‘4) Q = l + GT 0 o for scattering from the potential V0. Then the expression for T becomes T = (1 + T G)-1 T + V + (1 + T G)-1 T + v GT 0 o o o 1 1 = T + (1 + T G) v (1 + GT) 0 o 1 = T + 0 v 0 (3—5) where 00 = l + TOG is the time-reversed (incoming) solution correspond- ing to no. This relation is exact and involves no approximations. How- ever, it still involves the unknown scattered wave 0, so we still have an integral equation for T analogous to that defined in equation (3-1). If we expand out 0 we obtain that 0 = 0 + GV (0 - 0 ) + GV n . (3-6) 0 o o 1 Under the condition that Vo >> V1, 9 will be well approximated by 00 and we can drOp the higher order terms in equation (3-6). Then we get T . T + 0 V 0 (3-7) 0 o l 0 which defines the Distorted Wave Born Approximation (DWBA). SinceSlo includes the effects of V0 (the stronger part of the total potential V) 51 to all orders, this is a much better approximation than the PWBA de- fined by equation (3-3). For our applications to inelastic scattering, V0 is chosen to be the elastic scattering part of V, so that V1 then includes the inter- action that excites the inelastic transition of interest. We can identify V0 with the U0 obtained in Chapter 2 since they both satisfy the same integral equation (2-16 or 3-4). Since To does not contribute to inelastic scattering, the formula for T simplifies to be Tab =<¢b fiolvllflo ¢a> = ”—8) where x8 = 90 ¢a is the elastically scattered distorted wave which is the solution to the Hamiltonian Ho + V0. At this point it is easy to identify three limitations on the cal- culations performed this way that are essentially beyond our control. (1) The Optical potential used for V0 is necessarily approximate since the elastic scattering of pions is not fully understood. It is presumed here that the choice is reasonable if the elastic scattering is fit by V0. This will be examined in later chapters. (ii) The interaction V1 is not completely known, and we will include only the most important teams that should contribute to inelastic scattering. (iii) The assump- tion that V1 is small may not be satisfied, particularly in the case where coupled channels effects may be important. With these caveats in mind, we can now proceed to the evaluation of the formulae that are realized in the scattering program MSUDWPI [Carr] for use in performing the calculation which will appear later in this thesis. 52 3.2 REDUCTION OF THE CROSS SECTION FORMULA The T-matrix is defined in DWBA [Aus 70] as DW 3 3 (-)* + (+) + _ T zjfdrijflrf Xf (rf) X1 (r1) (3 9) where”? is the Jacobian for the transformation to relative coordinates (omitted from here on) and where = is integrated over the internal coordinates of the scattered particle wave function (W) and the nuclear state wave function (¢). This leaves <1'M'lV1|m> where I, M (I'M') and the initial (final) state spin and Z-projection. After a partial wave decomposition we obtain DW (-)* «y. A T I: If xfzvmvu ) Yz'm'(kf) 2m l'm' (3-10) (+) ‘* * ~ 3 3 0 I . I x <1 M lvllm> x]l (1') 12m (k1) dr dr 1m "" A where X£m(r) = ii (4fl)[hg(r)/r]ng(r) and where u£(r) has the normaliza- tion that Eim u (r) a-i sin (kr --&l - n ln(2kr) + 6 + 0 e161 (3-11) 1 k 2 2 2 ' I941) The formula (3-10) can be rewritten to give I)“ - ' ' ' 'l I A * A 1' Z ; <1 m 1 M JMJ) (1111 IM JMJ Y£,m'(kf) Ymuci) 9. ,T . J (3-12) + JMj + JM 3 3 1]]f [xz, (gm IMP?!) (111111 IMlJlMJl> (315) * A A x 1"m11MI'M'IJ >1: (k)Y,.(fc)Y (My. (1:) < J. 1 2m 1 g m f 21ml 21ml f This can be separated into two pieces. We first use the sum or M(M') to simplify the clebsches using triangular relations. Then the first piece is 54 * . . gull <9.m 1 M Jll-m|JMJ><1 m I MJ -m lmlJlMJDYmmi) Yfilmlflci) (3-16) Using that Y:m = Y£_m(-)m and then redefining m in the sum we get EL; <9” -m I M J+m1JMJ> (“111“ I MJ m11'J1MJ1>(')m Ymdéi) Y2 m (121) l 1 (3-17) These two clebsches can be rewritten using 3 J-1+m =- — - +m <1-m I MJ+m|JMJ> 1 ( ) <1m mJ’I MJ > and (3-18) 31 21+21'J +M ;m J M I __ — ‘ J -M o 21ml I MJ-ml‘ 1 J ~ ( ) <1 MJ In1 1 J llfil “11> 1 1 £1 1 Since I M +m I MJ J -M 1 - <1m JMj’ J >< -ml 1 J1. lm1> =ZL J-HL -I-L ; IL) (-) 1 Will IJ ; flJ) , (3-19) x IL W(£J£ J 1 1 1 we obtain the result that equation (3-17) is now 3 31 i ZJ'Jl'L+MJ1-m1 . . Z Z (-) Y1m<1m L (M-Hnl)lzl m1> (ILL ml, 11 ) 55 Since A <1m L-(ml‘Hn) '11-'15) =-:l (_)2-m <1m 11mllL m+ml> , (3-21) we can write equation (3-20) as 2J-Jl+2-L+M A A _ l o - .- Z J Jl( ) W(5LL 1J1, zlJ) 1 L (3—22) m+ml A A x: (-) (m 11mllL m+m1> Y“1 (k1) Y, m (k1) . m m l l 1 We can now use the result that Z <2m 11m1|LM> 12mm) Y1 m (k) m 1 1 1 (3-23) iii a 1“ HM YLM(k)<(9.O £10‘L0)> 41 L when M = m1+m2 is held fixed. Since M is not fixed in the sum above, we get a sum on M and the final result for equation (3—16) is given by “ ZJ-J +1-L+MJ “ "1 111 1 Z .______ (-) 1 NULL 1J1; llJ) (3-24) x <20 110|L0> YLMOCi) 56 The second factor that we can remove from equation (3—15) is " ' ' Z <£m IMJ-mlJMJ> 1 1 2m 1111 (3‘25) x<£imiI'JlM-mllM|J Jm,>Y' (EQY'm (12) J1 2 21m' 1 f In a derivation that follows the same pattern as we used for equa- tion (3-16), we rewrite this using *__m'* A Y£,m,(k) - < ) Y£,_m,(k) and reorder the clebsches. When the clebsches are combined into a Racah coefficient we get A +_' _! 1L, 2J Jl L +MJ ml 2 Z;- -—-—--<‘) l W(9.'L' I'J13 SLIJ) L m m1 (3—26) x<2mL'-(M+m)l 1“»th (inimiuc) which is analogous to equation (3-20). Then the sum on m and ml is used to obtain 1,... +-'+'+ J J1“ £1 2J J1 L 2 MJ1 2E: (- ) W(£'L'I'Jl; ilJ) L'M' F” (3-27) * A _ 1 1 1 1 k x 2;,ITI -z z 2 ml, 1.11 W1 LM L' 1M' JIM; 1 x <£0110[L0> <2' 012.10!L'O> W(2L 1J1; zlJ) x 1401’ L' 1' J1; 2'J) aLL, 5m, , MM JJ1 A <20 210|L0> - -E‘— (-)£ (20 L0|210>, £1 . . i 1' . . <11 0 £10|L0> - :— (-) <11 0 Lo|210>, ' 9* 1 and A * A 22 Y (k ) Y (k ) --E— p (cos 9) LM 1 LM f 4n L M where e is the (scattering) angle between if and Qi, to reduce equa- tion (3-28) to 58 A2A2 A2 «A, J J L 22 Z|I|2-Z Z Z 1 2 /2O L0l20> <2'o LOISLO) (4w) \ l 1 mm' L gg' J 9 “121 J1 J* J1 o l ! 1 . I x W(2L 1J1, zlJ) w <2 m I M lJMJ> w . “'M' (3-35) K At IN T I'M' Y x dg dg' Zézml nN < IAOA,u| > Au I z'm' . Au The second half of this equation can be further simplified to give 6O _ (-)* v (+) i l u r ii u2,(r) 2( > v t _________ At t v Z<2m qulz m) r 1TN Au Where the fact that the angular integral can be brought through tnN for (3-36) the interactions used here is proven in Appendix D. At this point it is convenient to define a new Operator u = (-) Tko (3-37) TAOk.-u A.u Then equation (3-36) becomes 2 W Mm A“ (3—38) (-)* v (+) u£.(r) 1 2 u2(r) _ 12 2 —-—————. 3 [X r [Arm <1“ TWA” I) r r dr and equation (3-35) then includes a sum on four clebsches Z Z(-)“ 11 mM m'M' (3-39) <2'm' Au|£M> in addition to the integral and the reduced matrix elements. The sum in equation (3-39) can be reduced to a single Racah coefficient. The procedure is to rewrite (I'M' A - u|m> -%'. (-))‘-” <2m'+p IM IJM> t J J mi <‘m' I' MJ-m'lJMJ> H)IH> =-{— (-)A if‘ wu'x JI; 91') I! - £319)ka mm' 11'; AJ) (3—40) which is independent of NJ. Substituting this into equation (3-35) gives 7112,? -Z ii (-)J-2-I mm' 11'; M) <1||YAHIU> A (3—41) 1’“ (I): 12' (+2 ) u r _ u , r xf—;L— Aan <1” TACK“ 1') rz r2 dr . The final step is to evaluate the reduced matrix element and put the equations into a standard form. First we use <9." YA” 2'> =- (410-1/2 <-)Ai <20 AO|£O> and - i' I-I' - > ' -_ - I <1" Txox ”1) i ( ) <1 H TAOA” I and (+) (-)* u2(r) = u£(r) P.- '5- a: 62 to get 212'1' = <4n>'1/ZZ(-)J+ZI+I' it mm' m; m x (”)2 12'-2< 20 loll”) (_)A-—I+I' (- )* (+) XI“ ”(1.) [MW (I'll TAOA”1>]I u (r) 2 dr ' (3—42) This can be more conveniently written as J _ -l/2 _ J+21+I'A, , . Tuz'I' (4n) Z< ) I W(I 1' 12, JA) Fun A where '... A u. + ' A Fum = (-)£ 12 9‘ 2 <20 Ao|2'o> (-)A I I A (3—43) * u u 1' - 2 2 xfT FA(r) -— r dr and Ekm = “we.“ EAOA" I> Thus we have an expression for TIIR'I' that separates the rotational prOperties of the matrix elements from the physics contained in th and«. Although the formula written here is restricted to natural parity transitions, it is modified in a straightforward way for unnatural parity (s-l) transitions. These details, along with those for normal parity transitions, follow in the next sections. 63 3.4 EVALUATION OF THE FORM FACTOR FOR NATURAL PARITY TRANSITIONS The form factor §A(r) defined in equation (3-43) can be evaluated for a number of different cases; we will examine only a few of them here. First, the transition density will be evaluated in the collective and microscopic models. Then it will be shown how the Impulse Approximation (IA) form of th’ as defined in equation (2-43), can be used to form §A(r) in several different ways. Finally, an ansatz will be shown for including the second order terms of equa- tion (2-55) in §A(r). A collective model transition density is particularly convenient for comparison to scattering by other probes because of its common use and simple definition. However, its use is best limited to those (usually low lying) states in nuclei where rotational or vibrational degrees of freedom dominate. This model assumes the nucleus behaves like a quantized liquid dr0p where the surface is deformed from a spher- ical shape to R(e,¢) - R°[l +2 (-)M EL_M YLM mm] (3-44) LM where - M “-1 M+ aL-M ()aLM‘BLL [bm+() bL-M] includes the Operators bLM (bifi) which destroy (create) an excitation with angular momentum L and the amplitude of the excitation BL. This 64 deformation produces a change in the density from a spherical form o(r) to C(rae:¢) a C(r) + (50 (3-45) - _ 39(1‘) Mr) Z OLLM Ro 3r YLM (6’¢) LM in lowest order. This deformed part is assumed to give rise to the nuclear transition. In this case (1' “TJOJ "1) = FC(r) = (01> II a. |l<01> > '._. .. .- (-)I J I v(Jo I'I; JI) HIM-I ' (HMO) M (') A (‘91 BJ J-l “-1 + >_ .-1 BJJ (3-50) * ug, X-j’ r flN which is the standard formula included in the original version of DWPI. 0 3r u ‘R fig)‘;&r2 dr A microscOpic model for the transition density has the advantage of allowing direct calculation of transitions between states which can be described by simple configurations in the nuclear shell model. The equations written here will be specialized to the case of transitions from a closed shell ground state to an excited state made up of a par- ticle and a hole coupled to good J. Such a state can be written as “14>. Z C13 Z since a hole state lhjm> a (_)j~m ajm|€> has quantum numbers jh and 'mjh° 66 The Cij can be obtained by diagonalizing a shell model Hamiltonian for the residual interaction between pairs of particle-hole states. The core C is assumed to be a filled shell. Such a process is known as the Tamm-Danckoff Approximation (TDA). A better approximation, one that accounts partially for the effects of ground state correlations, is the Random Phase Approximation (RPA). Here the ground state is assumed to include 2p-2h (and 4p-4h, etc.) components, so that a lp-lh state can be reached by annihilating a particle-hole pair (with ampli- tude Yij) as well as by creating one (with amplitude Xij)° This case can be treated in exactly the same way, where we replace Cij with X11 + (-)S Yij [Pet 70]. What remains is to evaluate the reduced matrix elements of s . TLSJ,M Z (In SA’JMJ> YLM oA , (3 52) J MA where 00-1 and 01:3, for a transition to the state in equation (3-51). Although only 8'0 transfers are considered here, it is convenient to also do the Sal transfer that will be used in Section 3.5. The par- ticle-hole matrix element is 'Z Z 2;; Cij ij m 1’1 3113-th ) < > x (- j 111 IT ’3 R (r R r 5J1. Rik) Rj(r) . 13 (3-53) 67 where Ri(r) is the radial part of the shell model wave function. A convenient formula for these is 1/2 n+£+l 2 2 2 (n-l)! ] a3/2 (ar)£e a r /2 -1/4 Rh2(r) . W [(2n + 22 - 1):! Pn2<-:- a) 1 1 (1112.1, £1£2L, 2 3 s) (3 57) S o A A J.""A _ j25‘23‘71‘S X 4r The results in equations (3-53), (3-56) and (3-57) can.be combined to give the result that fix“) =- At" FM(r) (3-58) N 68 where f 2 2 o 3 N -a r a — C FM(r) A o E N(Ctr) e N 2 2 f 4r = To- 0.3 |:A(0Lr)A + Mar)“.2 + D(or))\+4 + "1 e where CN includes the Cij: the reduced matrix elements, and the coef- ficients of the polynomial, and where N runs from the smallest value of 21+22 to the largest value of 21+22 + 2(n1+n2) for the configurations involved. FM is divided by A (the number of nucleons) to correct for the A included explicitly, and the normalization fo is used to correct for the extraneous spin factor of\/2-which is often included in micro- scOpic form factors calculated for proton scattering. The next step is to include the pion force for a normal parity transition. In the notation of equation (2.43) this takes the form _ -)’ Fx(r) =- 3§[A1F(r) + {75 A2F(r) v + v2 A F(r) (3-59) 2w 3 ] for F(r) = Fc(r) or FM(r). Notice that when Fc(r) is used this is just 3U ' a- _2P_£ - FA(r) Ro 3r (3 60) where the first order form of UOpt is used. If we make the ansatz that the form factor can be calculated in the same way (i.e., by differenti- ating the potential) from the second order form of UOpt: equation (2.55), then the effects of second order terms in tflN can be included by using 69 M (A +2A6 o) Fc(r) Efir) - -: [A1+2A4 p(r)] Fc(r) + I; 2 2m A [1+3 + E7- [2A5 p(r) FC(r)] E7 + v2 (A3+2A7p) ECU);
1 - T (:2) (3-64) n LlJ which differs substantially from the T (t) form that occurs (p)°T JOJ JOJ in normal parity transitions. Actually, the target space part is easy, and it has already been evaluated to give the result above in equa- tion (3-57). What is new is the current Operator 1 2J+l PJJ ”[T x PJJ -‘; JJ (3 65) ,-lJ J a [YJ-l® RF whiCh changes the angular momentum algebra in the derivation of the where £> T-matrix in Section 3.3. Specifically, the evaluation of <2'HYA in equation (3-38) must be replaced with the evaluation of 0 <9. " YJ_1.’J ” 51>. This matrix element is 3,12QO JMle'm'> m > MJmmU x (-)"<:5L'm"YLM Ijzm> . (3-66) [YL ® 9.] m" 2m> 71 + A 2+m+ 11 Since 2-u|2m> = \J£(£+l -% (“) <2m-u £-m|l-d> l2m-U> this becomes Z w mm) WJE(E;I3 H>|2<>> mm‘” (2' H Yr. "2) The sum on four clebsches can be reduced to a Racah coefficient where Z <2m JMJl2'm'> (-)mu m <2m-p 2-mll-p> <2m-p LMISL'm'> L+l-J+2 = i (-) 23 w(212'L; 2J) (3—68) 9. which can be used to get the result . , -l/2 31: 22 _ 2'-2+1-J <2||gml|2> (410 2' \/2<2+1) <> (3-69) x W(212'L; 52.!) £0 L0I2'0> . Since = (4H)‘1/2 23/2' (-)’L-Q"+J 20 JOIE'O> for non-spin flip cases, we see that the spin-flip term requires an extra factor of £2 1/2(2+1) w<212'L; 2A) ‘ (3-70) where L . A-1, and we also have to change (_)2 <20 20'2'o> + (-)L <20 L0I2'0> , (3-71) 72 which also changes the combinations of incoming and outgoing partial waves that match up in the integrals that form the matrix element. What remains is the evaluation of the form factor, which is - 2 S . 130:) = 1m X "n gLsmn) pJ J J <3-72) n where p: is given in equation (E-lS), and (he)2 28 gLS(q) = -4w (3 + s g-:) q (3-73) 0 l which is Obtained from equation (E-ll). This form factor in equa- tion (3-72) is calculated by ALLWRLD [Car 81a] and supplied to MSUDWPI for the scattering calculation. 3.6 SUMMARY In this chapter we have outlined the basic equations of the DWBA and their specific application to the problem of pion inelastic scat- tering. In equations (3-29) and (3-32) the formula for the inelastic cross section was presented in terms of a set of transition matrix elements. In equation (3—43) these matrix elements were expressed in terms of the overlap between the distorted waves and the pion-nucleus form factor for normal parity transitions. The modifications required for unnatural parity transitions were given in equations (3-70) and (3-71). Finally, the models for including the nuclear structure and the pion-nucleus interaction in the form factor were presented. The 73 collective and microscOpic transition densities were defined in equa- tions (3-49) and (3-58), while the form of the interactions to be inves- tigated were presented in equations (3-59), (3-61), (3-62) and (3-72). The remainder of this thesis will be concerned with the applica- tion of these formulae and the testing of these models. At first the pion-nucleus interaction will be tested against transition densities determined by nucleon scattering. Later, the knowledge of the inter- action will allow the study Of some new states and their nuclear structure. CHAPTER 4 INELASTIC SCATTERING IN THE COLLECTIVE MODEL The remainder of this thesis will be devoted to the examination of various test cases that illustrate the properties of the pion-nucleus interaction and the application of these techniques to the study of the nature of nuclear excited states. This chapter primarily addresses the first of these concerns: the following sections will examine transitions to collective states in order to study the properties of the pion-nucleus effective interaction. The existing collection of inelastic data place narrow limits on the choice of test cases. The nuclei for which data are available at a range of energies are 120, the 4.44 MeV 2+ state, and 40Ca, the 3.74 MeV 3' state. The former will be used in the detailed comparisons of the next three sections, while the latter will be included at the end for comparisons with microsc0pic models in the next chapter. The first two sections will examine in detail the effects of in- cluding various corrections to the pionrnucleus interaction at 50 MeV (Section 4.1) and 162 MeV (Section 4.2). The elastic scattering Optical potential will be fixed at one that fits the elastic scattering data, normally Set D defined in Chapter 2. The effects of changes in the distorted waves due to the use of other Optical potentials will be dis- cussed separately in Section 4.3. The remainder of the chapter will present a survey of low (Section 4.4) and higher (Section 4.5) energy scattering to states described by the collective model. This will review most of the states for which data exist, as well as some for which data is not yet available. 74 75 Throughout this chapter the curves drawn on the figures will follow a consistent pattern. A curve based on a theoretical force will be dashed and a curve based on a fitted force will be a solid line. A dash-dot curve will be used for a second fitted force. Exceptions to this convention will be noted as needed. 4.1 EXAMPLE OF COLLECTIVE MODEL AT LOW ENERGY The 4.44 MeV 2+ state in 12C has been chosen for this example because it is the best known angular distribution for low energy pion scattering. The other transitions that have been Observed at low energy are fairly recently studied and Often incompletely known. This state offers the additional advantage of having been extensively studied with other projectiles. The use of a form factor determined by an indepen- dent experiment allows the separate investigation of the prOperties of the pion interaction. For these calculations we will use the collective model described in Section 3.4, with 82 = 0.60 taken from proton scattering analyses [Fri 65]. The elastic scattering Optical potential will be the one given by Set D in Table 2-4. The only remaining factor in the model is then the pion-nucleon interaction, which will be examined by comparing the predictions of the various interactions presented in Chapter 2. The effects will be illustrated by plotting the pion-nucleus t-matrix, using the local approximation described in Appendix B, alongside the angular distribution [Dyt 79] for the inelastic scattering. The figures will also show the elastic scattering data taken by the same experimental group, for scale. The other elastic data are from [Pre 81]. 76 The first calculations use the four parameter potential develOped in Section 2.2 based on the Impulse Approximation (IA). The IA potential using phase shift parameters (Set A) is shown as the dashed curve in Figure 4-1. The minimum in the angular distribution occurs at 80° rather than at 65° as required by the data. This is the same shift that we saw for the elastic scattering earlier. The plot of the t-matrix on the right shows quite clearly that this shift is due to a defect in the force, specifically the relative strengths of the s- and p-wave param- eters. The solid curve is the result of the fit to all four parameters, Set B. Notice that when the elastic is fit the interference minimum falls in the correct location. The dash-dot curve uses Set B', which only varies three parameters in the fit. This alternate fit is quite similar in character to the other fit -- leading to the observation that forces with quite different parameters can be essentially equivalent in their description of the pion-nucleus interaction. Figure 4-2 shows the same case, except that the full potential was used. The dashed curve is the purely theoretical potential based on phase shift values for the parameters, with absorption parameters as calculated by Riska [Cha 79a]. Agreement with the data is quite good. The graph of the t-matrix shows how the LLEE effect and second-order s-wave parameters give the correct interference. The solid curve shows the fitted Set (D), where the only major change is a further increase in bo by 502. This shifts the minimum a small amount and raises the inelastic calculation to give slightly better agreement with the data. It has been noted before [Str 79, Dyt 79] that potentials which fit the elastic will also reproduce the inelastic data. Now we see a 77 .ham>fiuomdmou .mm>uso uoclnmmv was vfiaom .vmnmmw nous macaw ..m can m .4 muom aouw muouoamumo cows umsawmw Aunmau one so vmuuoaov xquumalu wofiumuumom Oaummamcu cm can a umm Boom muoumamuma Hauauoo magma mumum A+~v >0: qe.< mufi can UNH scum +e >02 on no wawuouumom owummHoaH new OaummHm Hus «Lamas UX -8|—099 \J 78 In TIII f II\III T r IIIIIIYI I [I'IIII I U A I— \ .1 1— -1 L _o. g r—\ H .. “ l . 4 E :3 L- -I .— 4 O In L‘ as b d - -I P -I )- -I o 111111 1 1 1111L11 i 1 lLLLiLl 1 1 l1111l1 1 1 c5 3" m N C) H O C) O o—c .—a .-o '4 [suun -qu] I )0 epmzufiow IIITII I I [1111]! I I [IIITrIn I [111111 I I O “[0 H d do (\1 H A d _C> 07 _C) (D .7 d _O m fi 11111 1 [1111111 1 i1111111n 11111111 1 O o o 0—0 v—0 0 Q .-I . O o—o O. H [Js/qw] 6p/Dp Figure 4-1 GCJnldeg] 79 .>Ho>«oooomou .mo>uao uaaom can cocoon nua3 c3056 .a can 0 name scum mumumamuma suHB umumazoamu was xuuumalu one umooxm .Hle muswwm mm meow mus «unwed MSUX~81~|OO 80 IIIII I I I [III I I I I IIIII] I I I IIIII] I I I 3 I— ‘0. HF“ :— 1‘ Q—. I u. b- q ’ . 0" _ ‘10. o - o 1111 1 111111114111111111 1 11111111 1 o' :r m N O o—o Q C) C) -—I F. F. H [suun °qu] 1 )0 apnuufiow IIIIIII I [IIIIIII I IIIIIIII I [IIIIIII I q (3 “[0 H O _N 0—4 _0 0') q .10 (0 q do m 1111 1 11111111 1 11111111 1 11111111 1 O v-o 0. v—O [JS/qw] tsp/op Figure 4-2 9c.m.[deg] 81 much more interesting result: potentials that give similar fits to the elastic data will also fit the inelastic data, even if the Optical potential that generates the distorted waves is held constant. The fact that a four-parameter and nine-parameter potential give similar results suggests that they are both modeling an effective interaction that describes the pion in a nucleus. Thus we have the advantage of using the full potential when discussing the values of the parameters in the context of multiple scattering theory, and also using a-completely equivalent four-parameter potential to discuss the qualitative behaviour in a simple and straightforward fashion. In summary, the 50 MeV inelastic data for 120 are well described by any of the potentials (B, B', D) which were fit to the elastic scat- tering. The theoretical Set (C) also does an adequate job. This can be attributed to the similarity of the t-matrices for these potentials, which reflect a common effective interaction. 4.2 EXAMPLE OF COLLECTIVE MODEL NEAR THE RESONANCE The same state (4.44 MeV 2+ in 12C) and transition density (col- lective model with 82 = 0.60) will be used in this example. The data are at 162 MeV [Cha 79] for both w+ and I". We will only look at the fl+ here, but both will be shown later in Section 4.5. These data are particularly interesting because they were taken all the way back to 180°, the only such case currently available. The elastic scattering is calculated using Set D for every case. The first set of calculations, shown in Figure 4-3, use the IA form of the force given in equation (2-43). The dashed curve shows Set A, which uses the phase shift parameters with no adjustments. It 82 .m~m>«uomqmou .mm>uso uOUInmmv new vwaom .umcmmv sufiS naonm ..m can m .< mumm aouw muoumamumm no“: vocwmmv Aunwfiu mnu so vmuuoaav xfiuumalu wcfiuouumom ufiummaosq cm can a umm aouw mumumamumo Hmuauao mafia: oumum A+Nv >mz q¢.¢ muH new DNA Eoum +: >mz NQH mo mcfiumuumom afiummawcw can OHummHm mns magmas MSUX-Bl - 101 83 IIIIIITII\ IIIII[I I I IrrIIIIIfi [IIIIII I I w I— "N P‘ .-4 I + ‘ L” 0“ II- d” 11111111 [1111111 1 [1111111 1 [1111111 1 O 3" m N Q d C O O H -—4 o—l 0-0 I ('0 [suun qu] I )0 epnuufiow I c OJ ‘5 IIIIITI I [IIIIIII I IIIIIIII I IIIIIIII I IIIIIIII I IIIIIIII I [IIIITI Ij .240 [.11. FR 0') CD '0 H. E. 0 <1) .4. o ca IJS/qw) asp/op 84 does a fair job of reproducing the experimental values at small angles, but it has the wrong phase. This is the same problem we had with the 162 MeV elastic data. Notice that the angular distribution is now diffractive, with the minimum in the force at 70° not evident in the calculation which has a minimum at 65°. The solid curve uses Set B, which was fit to the elastic, but there is little improvement. The dash-dot curve used Set B', where the v2 term was drOpped, and the re- sult is an improvement in the first minimum at the expense of the back angle data. The t-matrix shows the shift of the minimum to larger q, so the nucleus looks smaller and the minimum is shifted. Notice that the second minimum is not affected, and thus the apparent improvement is not really very great. Figure 4-4 shows the same series of calculations, but the full interaction given in equation (2-55) has been used. The difference between the theory (Set C, dashed) and fitted (Set D, solid) curves is negligible. Except for the phase at the first minimum and the rise at 180°, the data is well described by this calculation. As before, dropping the V2 terms (Set D') improves the fit at the first minimum but fails at larger angles. The pion-nucleus effective interaction is fairly well described by the full potential with either theoretical values (phase shifts and Riska's absorption values) or adjusted p-wave strengths. The four- parameter model does not seem to be completely equivalent. It is similar at forward angles but lacks the density dependence that seems important to the description of data between 90° and 180°. Although Set D fails to reproduce some important details, there is a dramatic improvement at backward angles. 85 .>Ho>wuooommu .mo>u=o uoulsmmv cam cfiaow .wonmmv cows SBOLm ..a can a .0 mumm scum mumumamuma nufia vmumasoamo mos xfiuumalu mnu unooxo .mlq muawfim mm meow one madman MSUX-81-102 86 \ IIIIIII I I IIIIIII I I \ IIIrIII I n 3‘! 1—0 huLLL4_JMMlLL4.4I . 1 111111 1 1 111114 1 1111111 1 1 1111111 1 1 Q 3' m N O 0.4 C) O O H H —I u-4 [sIIun 'qu] I I0 epnIIufiow IIIIII I I [IIIIIII I IIIIIII I I IIIIIII I I [IIIIIrI I IIIIIII I I [IIITIII T .. I j .100 q _ (\1 31 O .. H J?‘ n h .. d -—0 .1 _O 0') .0 L0 —1 _.O __ (T) Llllllll I Inn“. 1 llllllll y; Im1111 1 11111111 1 ' ' o o . O O c—I 0 —¢ 8 H IJS/qw] esp/op .01 .001 Figure 4-4 QCJnldegl 87 In summary, the calculations with the full potential describe the data much better than the other alternatives, particularly at back angles. If the phase problem near 70° can be understood, then large angle data will become a crucial test of the form to be used for the force and any possible coupled channels effects. 4.3 DISTORTED WAVE EFFECTS The calculations in the preceding sections used a fixed optical parameter set in order to eliminate any variation due to changes in the distorted waves. However, the choice of Optical parameters is not well determined and it is important to identify how these changes affect the inelastic cross section. Figure 4-5 shows ISRIZ, where S2 = exp (2i52) and 6£= phase shift for the 2th partial wave. The magnitude of S, is a measure of the transparency of the potential, since it is 1.0 when the phase shifts are purely real and less than 1.0 as flux is removed from that partial wave. As can be readily seen in the figure, there are some significant differences in the makeup of distorted waves pro- duced by different potentials. The 50 MeV curves are for sets A (dashed), B (solid) and B' (dash- dot). The different absorption required by the fitted sets is clearly evident; what is surprising is the large differences between the two fitted sets which reflects the different distribution of the absorption between the s- and p-waves. The 163 MeV curves show little of this sensitivity. Sets A, B and B' are all strongly absorbing and determine the same radius, so different choices should not affect the inelastic scattering results as much as they do at 50 MeV. One can also see why a diffraction model, which assumes a sharp cutoff at a characteristic 88 Figure 4-5 Plots of [Sglz for elastic scattering of 50 and 162 MeV n+ with parameters from sets A, B and B', shown with dashed, solid and dash-dot curves, respectively. 89 MSUX-BHCB I I I I 1 I III I r ” 50 MeV * 0.0 1 1, 1 1 L 1 1 1 AL 0 5 10 |_ I I I I I I r’ r I 1 10 Figure 4—5 90 value Of 2, would give a good approximation to the resonance region data. The effects of distortion will be illustrated by the same calcu- lations as we used in Sections 4.1 and 4.2 except that now the t-matrix used in evaluating the form factor is held constant (at Set D) while the Optical potential is varied. Thus the figures will now have pairs of curves, the first showing the change in the elastic scattering cross section and the second showing the induced effect on the inelastic cal- culation. The 50 MeV results will be presented first, followed by the 163 MeV calculations. The solid curves in Figure 4-6 use Set D for the elastic, and are thus exactly the same calculations as were shown in Figure 4-2. The dashed curves show the effect of using Set C, the theoretical values, in the full Optical potential. The curves are slightly higher, but clearly the change in bo does not have a big effect here. The curves in Figure 4-7 show a much greater sensitivity to the potential. These use the four-parameter potential with sets B (solid), 3' (dash-dot) and A (dash), which were the ones used in Figure 4-5. Using Set A destroys the agreement in the inelastic calculation, as might be expected. What is more interesting is that sets B and 8' pro- duce measurably different results (as Figure 4-5 would suggest) while the elastic calculations differ mainly in the coulomb-nuclear inter- ference region. It's clear that measurements in this region would help pin down the Optical potential at low energy, although data taking is difficult at these angles. These results are summarized in Figure 4-8, where both the optical 'potential and the inelastic scattering t-matrix are varied. The solid 91 Figure 4-6 Elastic and inelastic scattering of 50 MeV n+ from 12C and its 4.44 MeV (2*) state using Set D to calculate the inelastic scattering t-matrix, while the Optical potential used parameters from sets C (dashed curve) and D (solid curve). do/dQ [mb/sr‘] 100. 10. H o .01 92 MSUX-Bl-IO4 IIIIIIII I IIIIIIII I IIIIIIII I IIIII 1 1 1 J 1 111111 1 1 1 111111 1 1 1 111111 1 1 1 11111 1 I 90 ec.m.[d99] Figure 4-6 1 l 1 1 1 1 120 150 93 Figure 4-7 Same as Figure 4-6, except the Optical potential used parameters from sets A, B and B', shown with dashed, solid and dash-dot curves, respectively. dU/dQ [mb/sr‘] 100. 10. 0—5 o .01 94 MSUX-8l-IO5 IIIIIIII I I lle 1 1 111111 11 1 1 111111 1 1 1 111111 1 1 1 11111 1 1 l 1 1 J 1 1 I 90 120 150 ec.m.[d°9] Figure 4—7 95 Figure 4-8 Same as Figure 4-6, except the same parameters were used for both the optical potential and the inelastic scattering t-matrix, either Set B (dashed curve) or Set D (solid curve). do/dQ [mb/sr] 100. 10. H o .01 96 MSUX-Bl - |O6 I I’rIII I I II’IIII I I I II III] I I I ll IIII I I 1 l 1 1 111111 1 1 1 11111 1 1 1 1 111111 1 1 11111111 1 l 90 ec.m.[d99] Figure 4-8 1 l 1 .1 l 1 120 150 97 curve uses Set D (fitted with full potential), while the dashed curve is Set B (fitted with four-parameter potential). We saw in Section 4.1 that these potentials produced equivalent inelastic cross sections, while in this section we saw that the results were extremely sensitive to some details of the Optical potential. In this case, most of the variation in the inelastic results is due to the change in the Optical potential. Thus at low energy we have a fairly well defined effective pion-nucleus interaction for inelastic scattering, but are quite sensi- tive to the choice of the distorting potential. We now turn our attention to the effects of distortion at 163 MeV. The t-matrix for the inelastic transition is fixed using the parameters of Set D. Then the solid curves in Figure 4-9, which use Set D for the elastic, are the same as in Figure 4-4 for comparison. Clearly it makes little difference whether we use Set D or sets C (dash) or D' (dash-dot) for the distortion -- the inelastic results are nearly identical. The same conclusion can be made from Figure 4-10, where sets B (solid) and B' (dash-dot) also give nearly identical results. The only exception is Set A (dashed), which is based on phase shifts only, where the in- elastic calculation comes out lower than the others. This last case corresponds to the dashed curve in Figure 4-5, which differs from the others with respect to the sIOpe in the surface region. These results are summarized in Figure 4-11, where both the optical potential and the t-matrix are varied. The solid curve is for Set D (fitted with full potential), while the dashed curve is for Set B' (fitted with four-parameter potential with no v2 term). We saw in Section 4.2 that these two sets produced very different inelastic cross sections when the same optical potential was used, whereas we just 98 Figure 4-9 Elastic and inelastic scattering Of 162 MeV fl+ from 12C and its 4.44 MeV (2*) state using Set D to calculate the inelastic scattering t-matrix, while the optical potential used parameters from sets C, D and D', shown with dashed, solid and dash-dot curves, respectively. do/dQ [mb/sr‘] .001 99 MSUX-Bl- 107 10. p_; (2) E3 . . l" P l I IIIIIIF I I IIIIrq-7’xLl'r1Trrq—r—I-I-rrnq—I—rr1'nn‘ y_a .1 g .0157 1L1 1 1 1 11111 1 1111nfl 1,1 1111H1 11111 111 1 1111ud 1 1,11111fl 90 ec.m.[degJ Figure 4-9 I 1 1 l 1 120 150 100 Figure 4-10 Same as Figure 4-9, except the Optical potential used parameters from sets A, B and B', shown with dashed, solid and dash-dot curves, respectively. dU/dQ [mb/sr] .001 100. 10. H O H IIIITIII'l—TIIIIIIII IIIIIIITF IIIIIIIII7WIIIIIIIII IIIIIIII H .01 lOl MSUX-8l-IOB 1 111nm 1 1111ud I. K 11 Md 1 1.1111u1 1 1 11111d .1 1 11111d 1, 1 1 111111 30 ec.m.[deg] Figure 4-10 120 102 Figure 4-11 Same as Figure 4-9, except the same parameters were used for both the Optical potential and the inelastic scattering t-matrix, either Set B' (dashed curve) or Set D (solid curve). do/dQ [mb/sr] 100. 10. 10. .01 .001 103 MSUX-Bl- 109 90 ec.m.[deg] Figure 4-11 1 1 1111ul 1.1 1111 1 1 1 111111 1 1 1111M] O 1 1111n1 1 1111nd 1 1111ufl l 1 1 1 1 l 1 120 1'50 104 learned that the Optical potentials alone do not affect the inelastic results in any significant way. The entire variation in the inelastic calculations is due to the change in the t-matrix used to calculate the form factor -- exactly the Opposite of the situation at low energy. Thus at high energy we have a reasonably well defined distorting potential (despite some rather Obvious defects) but greater sensitivity to the ef- fective interaction. The full potential, with some adjustments, seems to be preferred over the other models that have been examined. 4.4 OTHER LOW ENERGY CASES We will now proceed to survey the existing experimental data to see how their trends fit into the patterns that have been outlined for the two specific cases above. This section will only review data below 100 MeV. This division is convenient and historicalI the only apparatus for such experiments is the Low Energy Pion (LEP) channel at Los Alamos, and most data there were taken near 50 MeV. NOw that LEP has been pushed up to 80 MeV, this division is less sharply drawn. Indeed, one goal of current research is to predict the complicated transition be- tween the simple descriptions of 50 and 163 MeV scattering that have been given here. For simplicity, these discussions will be restricted to two param- eter sets. One will be Set C, which uses phase shift values [Row 78] for the single nucleon parameters and Riska's values [Cha 79a] for the absorption. These are given in Table 4-1 for the energies we will study in this section. The other will be Set D, which has Reba, Reco, and the amount of absorption (with ImBo/Imco fixed at the pionic atom value) adjusted to fit the elastic data. These are given in Table 4-2. The 105 Table 4-1 Parameter Set C Theory Values for Low Energy Scattering *Linearly Interpolated. 36 MeV 50 MeV 67 MeV 80 MeV -0.041 -0.045 -0.051 +0.055 +0.004 i +0.006 i +0.009 i +0.011 i -0.131 -0.131 -0.130 -0.129 -0.001 i -0.002 1 -0.002 i -0.002 i 0.71 0.75 0.79 0.82 +0.011 i +0.028 i +0.063 i +0.10 i 0.44 0.45 0.47 0.47 +0.005 i +0.013 i +0.031 1 +0.05 i 1.6 1.6 1.6 1.6 ~0.01 -0.02 -0.03 -0.04 +0.12 i +0.14 1 +0.16 1 +0.18 i 0.33 0.36 0.41 0.48 +0.50 1 +0.59 1 +0.74 1 +0.92 1 Table 4-2 Parameter Set D Fitted Values for Low Energy Scattering 36 Mev* 50 Mev 67 MeV 80 MeV +0.056 -0.060 -0.048 -0.026 +0.004 i +0.006 ' +0.009 ' +0.011 i -0.131 -0.131 -0.130 -O.129 -0.001 i -0.002 ' -0.002 ' -0.002 i 0.74 0.75 0.64 0.54 +0.011 i +0.028 ' +0.063 ' +0.10 ' 0.44 0.45 0.47 0.47 +0.005 i +0.013 ' +0.031 ‘ +0.05 ' 1.6 1.6 1.6 1.6 -0.01 -0.02 -0.03 -0.04 +0.14 1 +0.12 1 +0.13 1 +0.28 ‘ 0.33 0.36 0.41 0.48 +0.77 1 +0.66 1 +0.76 1 +1.54 ’ 106 one exception is at 36 MeV, where the values were interpolated between zero and 50 MeV since the data seemed too sparse for a reliable fit. There are indications [Car 81] that this procedure can explain all the data between zero and 50 MeV within systematic errors. Some conventions have been chosen. Set D is shown with a solid curve. The elastic and inelastic calculations use the same force. Re- sults for w‘ scattering have been shown whether data exists yet or not, since these are an important prediction of the model. Solid points are used for fl+ data, while Open ones indicate I” data. We begin with 36 MeV calculations for scattering from 12C (4.44 MeV 2+ state) and 28Si (1.73 MeV 2+ state). The density parameters are given in Appendix C. The collective model form factor for 12C uses 82 = 0.60 [Fri 65] as before.“ The form factor for 28Si uses 32 = 0.40 [Ful 68]. The results of these calculations are shown in Figure 4-12. The data are all from [Ama 81]. The results for II elastic scattering are fair, as might be ex- pected since Set D was not fit to the data. This set agrees much better [Car 81] with the 40 MeV [Ble 79] and 30 MeV [Pre 81] elastic data taken by a different experimental group on the same beam line. There must be some systematic normalization differences. Notice that the shape, particularly the absence of a clear dip in the coulomb-nuclear interference region near 30°, is well reproduced. The inelastic calcu- lations slightly overestimate the cross section for the 2+ in 12C, but underestimate that for the 2+ in 288i. The latter will turn out to be a persistent problem, and will be discussed at length in Section 4.5. The r‘ predictions are shown on the right. The characteristic shape change from W+ to fl‘, due to the change in sign of the coulomb 107 Figure 4-12 Elastic and inelastic scattering of 36 MeV w+ and w‘ from 120 (top row) and 2881 with parameters from Set C (dashed curve) and Set D (solid curve) as described in the text. 108 MSUX -8|-| IO :44..fi . 114~nq 4 114:4: q —.:.-qu 1 _ 1 fl 0 15 1 1m. l 10 9 10 8 L L 10 3 L _F» b _.:.p. p - b::p.h _ —.:-~ _ p O O I 1 1 m w 1 o o. 1 .co\nE_ Gu\bo «ddfiq d 1 —«~«.— 4 a ‘ cud-qqd « q fijqdfi d 4 1 + 1 .fl 0 15 1 l 1 1m l A 1 0 l I! 9 L / 1. 1 r0. 10 ‘ 5 T. 1 .Y 1 \ 10 3 IA Y 1 :hphb» - rppih - P pun-uh. . p 7:... p . O o o 1 1 0 0 1A . O 1 o. E}... %\8 ec.m.(d°9) o0.1111699l difidq 1 1! —:qqqq « q 114dd404 dd-dde A d n _::_L_ P —-:hhp . p 111 L 1 J l 80 90 120 150 30 rhpp—. » b o I 1 0 l o rose. .o..\8 .01 dzqudd d d .u-ddq4 1 d —uso uHHomv o uom paw Am>u=o conmmvv o uom scum muwumamuma magma UNH Scum I: mam += >oz me «o wcauwuumum ofiummamcfi can ouummam oaus shaman MSUX-8| -I I4 118 90 _ 9c.m.[d99] 60 30 IIIIIII I IIIIIIII I IIIITII I I [TIIIII I I ["1111 I I - l 4 / ._ / —I / — I q r- , -I /- Io-I | — .- +04 -J / o g-O—A - / I-o—‘I - O O I- /. I ‘4 l . . I— Q —1 \. I- ’ ’—£ ‘f \o \\ I— Q d I. ) I— /o .——c 7‘ 4 —I a. / - . l . O ./ \ 1 . \ .. \ .1 - \ 11111 1 1 [111111 L1 111111191 L illllLLl 1 1111111 1 1 I o o \\ o 0—4 u—c o O H H o O O v-O o H [JS/qw] esp/pp rIrIII I I [111]]! I I llIllll I I IIIIII I I I IIIIIII I I O _ H t --m c—I d O “N H .1 A .1 .1 c1 L11 1 1 J111111 1 1 l111111\\1 1 I11L1111 1 [111111 1 1 o o g \\ o v-i H C) Q o-I 1—1 . O O H - fl [49/qu asp/op 90 120 150 ec.m.[d°9) 80 30 Figure 4-16 0 I Is 119 data only. The consistency of the results is quite impressive. The elastic cross sections for n‘ are fit well by Set D. Both inelastic cross sections rise faster than the calculations until they are a factor of 2 higher at back angles. This deviation at high momentum transfer is not well understood. The authors of [Ama 81] have proposed that coupled channels effects may be important since the inelastic cross section actually dominates for 6‘: 130°. It was also preposed that phase-shift equivalent distorted waves that have very different in- terior forms can produce this effect [Kei 81]. One can summarize the ideas on this by saying that it could be due to the force, the reac- tion mechanism, the form factor or the distorting potential choice. Since this effect has appeared just as we enter a previously unstudied region, it is difficult to disentangle these possibilities at this time. Finally, we turn to the 80 MeV results. These are for 12C (4.44 MeV 2+ state), 400a (3.73 MeV 3‘ state), 90Zr (2.75 MeV 3' state) and 208Pb (2.62 MeV 3' state). The deformation parameters are the same as quoted earlier, except we now add 83 = 0.14 [Bin 73] for 902r. The n+ results are in Figure 4-17, while the n” results are shown in Fig- ure 4-18. The data are all from [Ble 81], and are preliminary at this time. The calculations are very interesting, which means they do not explain the data. The n+ elastic data show a curious trend from 12C, where Set D provides a good fit, to 208Pb where Set C is better. Although Set D was only fit to the first two lobes of the angular distribution, it is surprising to see it do so poorly since this same method works well at higher energy. Perhaps more surprising is that Set D reproduces the n‘ data quite well despite the fact it was fit to the fl+ data alone. 120 Figure 4-17 Elastic and inelastic scattering of 80 MeV fl+ from 120, 400a, 90Zr and 208Pb using parameters from Set C (dashed curve) and Set D (solid curve) as described in the text. 121 MSUX-Bl-IIS 1000. 0 qua—.1 d Edd-«u d 1.1“- : q 1d1d—qd d Eqdfiq 4 Edd—«d. ‘ C I 0 uI U l I} O 1 :.:p» p 53:. r rip-b? . 5:: 7x. rtlpp . _:::p p . 1 take. au\bo dddu d d d dfidddd d d 1 judddq d d id‘ddq d a. —ddd<-d .- a r C F 2 1 'l I. \ \ I \ 00 \ I \ . C I a o v / O . v' I \ \ ,l\ A :prp P r::-» . 72:2. . —::pb~ » r::.. p o o a II a 1 1 0 0 1 1 I o 0 1 o 1 EEE as}... 90 120 150 e¢:.1'I\.(d°9) 60 30 1 0 0 1 1 o 0 90 120 150 ecmwegl 80 30 11:. < q 1:.«4 . 1.4«qdi1 1::.. u 1u=u vwflomv a umm cam Am>uso monmmvv o uwm scum mpwumsmuma wcwm: «mam scam I: mam +=.>oz omH mo wcaumuumom ofiummaoaa mam owummam omlq ouswam MSUX-8l-ll8 131 "ITITV I llllllll I IIIIIII T T IIIIIIII l [UTIII 1 1 [Illllll I UIUIII I I IIITIFI 1 r F111]! 1 r IIIIIWT T "HUI! I I IIIUIII I I (7) (I) ”N r I 150 b 90 120 ec.m.(d°9] 60 30 1111 1 1 1111111 1 1 1111111 1 1 111111111“ 1111111 1 1 1111111 1 1 \ I I O Q \ Q H H O C) D H H 0 O o o H - 0 4-4 .—. [JG/qw] Gp/Dp ec.m.[d°9) Figure 4-20 132 and it would be necessary to develop the apprOpriate codes in order to perform the prOper analysis of data like this. We now return to the 162 MeV data that was examined previously. The calculations and data [Cha 79] are shown in Figure 4-21. The n+ results have been discussed extensively above, so this will not be re- peated here. The main observation is that the W‘ results look very much like the w+ ; they suffer from the same phase problem relative to the data. The overall agreement is good, with a suggestion that Set C may be better at back angles. There is a possibility of learning more about the density dependence of the interaction from large angle data of this kind. The final case is shown in Figure 4-22. These 180 MeV data are for 283i (1.77 MeV 2+ state) [Pre 79] and 400a (3.74 MeV 3’ state) [Mar 80]. The calculations are quite good, with the fitted set (Set D) improving the results in the minima as a result of the decrease in the absorption used. The inelastic data for 28Si are a factor of 2 above the calculations, as explained earlier. The results here look exactly like those at 130 MeV, which shows that the theory has the energy depen- dence under control in this region. The same is true of the 4oCa data, which is described as well here as it was at 116 MeV. In summary, the results of these calculations are quite good given that the theoretical underpinnings are poor at these energies. It is particularly interesting that the trends of the data support the use of Riska's absorption values, despite the sometimes unphysical assumptions that enter the calculation. These numbers and the 02 parameterization seem to provide a basic description of most of the data, elastic and 133 .uxmu mzu :H monauomwv mm Am>u=o vfiaomv a uom cam Ao>uso umnmmvv o umw scum mumumamumn wsfim: UNH aouw I: mam += >oz Nod mo wcaumuumom ouummaocfi mam oaummam Hale ouswfim MSUX-8l-ll9 134 h— _ 10 IIIIII I I 11111]! I I IIIHIII r [UNIV] 1 lmrrrf 1 IUIIIII I IIIIIIFI I I 0 O / H H o v—d H [JG/(1W) (Sp/DP illlllll l llllllll I Illlllll r 11111111 1 11111111 1 WM :5 o O ' .01 1 150 .001 9c.m.[d99] ec.m.[deg] Figure 4-21 135 Figure 4-22 Elastic and inelastic scattering of 180 MeV W* and n" from 28Si (tOp row) and 400a using Set C (dashed curve) and Set D (solid curve) as described in the text. 136 MSUX-81420 1000.1.11r11111vfirr11 1000.uvlva#1fT11rvrv 288; 100. 100. 10. 10. E 1. E 1. > > , E E _. 10. _. 10. 01 01 U U \ \ 8 1- g 1. .1 .1 .01 + .01 11 leLLlLlllllllJll 111L11111L111LJ11 '0‘” 30 so so 120 150 '0‘” 30 so so 120 150 ec.11‘1.[d°9] ecmw'gl lOOO-IFIIrIrr[II[tr[r 1000.1111rlvr1111111r 100. 100. 10. 10. E 1. E 1. > > E E H 10. '- 10. g g E \ ‘o 1' 8 l. .1 .1 .01 .01 1'1 11141111L11111411 '0‘” 30 so so 120 150 '001 30 60 90 120 15° oc.m.[d‘9) o1:.1'11.(d°91 Figure 4—22 137 inelastic, in this region. There are two areas of difficulty. First, the phase of the angular distributions at 162 MeV are incorrect, while those for neighboring energies are fine. Second, the calculation for the 28Si 2+ is always low, but this is likely due to a problem in getting the right value of BL for use in a direct (rather than coupled channels) calculation. Overall, it appears that the Optical potentials produce distortions apprOpriate for these energies, and the inelastic t-matrix is suitable for the analysis of these T=O transitions. 4.6 SUMMARY This chapter used the collective model for low-lying T=0 states to illustrate the properties of the pion-nucleus interaction and the effects of the distorting potential on inelastic calculations. A survey of the existing data was then used to gain a perspective on the quality of the fits as a function of energy, and the changes in the importance of various effects as the beam energy nears the A-resonance energy. In Section 4.1 the distorting potential was held fixed so that the properties of the low energy pion-nucleus interaction could be examined without other complications. It was found that a large range of inter- actions produced equivalent fits, so that it made sense to speak of an effective interaction at these energies. The interactions from elastic scattering fits and multiple scattering theory were similar because they all had enhanced s-wave repulsion and reduced p-wave attraction, as required by the data. The interaction near resonance was examined in Section 4.2. It was found that the form of the interaction became important, since the full potential produced better results than a four-parameter model which 138 also fit the elastic scattering. It appears that the density dependence is a major contributor to large angle results. The effects of distortion were examined in Section 4.3. The in- elastic calculations at low energy were found to be sensitive to the choice of Optical potential, so that more complete angular distributions would be needed to firmly establish the potential to be used for cal- culating the distorted waves. Figure 4-5 showed how the low energy potentials had varying amounts of transparency so that the results depended on what parts of the transition density were sampled by the scattered pion. At high energy the potentials all produced a "black" nucleus, and as a result the inelastic calculations were relatively insensitive to the choice of distorting potential. The remaining sections reviewed the existing experimental situa- tion. It was found that the low (ETr §_SO MeV) and high (E1r Z_llS MeV) energy data were well described by both the multiple scattering theory potential (Set C) and the fitted potential (Set D). The exceptions are the 2881 2+ state, where the difficulty is associated with the choice of BL, and 162 MeV scattering, where the calculations are consis- tently out of phase from the data. The data in the transition region (50 MeV 1E1T §_lOO MeV) were not as well described by these calculations. There was a problem at backward angles for the excitation of the 12C 2+ state with 67 MeV pions, but this may be specific to this state. The problems at 80 MeV were more general, since no interaction could fit all of the data. At this point the situation is not very conducive to extracting information about nuclear structure, but should be very useful in understanding the interaction in this region. CHAPTER 5 MICROSCOPIC MODELS FOR INELASTIC TRANSITIONS The previous chapter was limited to the consideration of the collec- tive model for the transition density in order to focus attention on the effects associated with various forms of the pion-nucleus interaction. The collective model is also widely used in the literature because the most commonly observed states are easily described in this way. However, states which can be described by a'simple shell-model wave function are best studied with microscOpic densities. It is this wider class of problems that motivates the examination of these densities. The first half of this chapter will continue the study of natural parity transitions. Section 5.1 will demonstrate a reasonable ansatz for including density dependent terms when using a microsc0pic density. It will also examine the effect of changes in the form factor on the calculations, which was omitted from Chapter 4. Section 5.2 will then examine some additional cases of interest. The second half of this chapter will examine unnatural parity transitions. Section 5.3 will outline the way these calculations are done, using the 6‘ T=l state of 2881 as an example. Section 5.4 will examine two other cases of in- terest at tests of the model. 5.1 COMPARISON OF COLLECTIVE AND MICROSCOPIC MODELS Before studying particular states, we will first check that we understand how to use a microscOpic model with the density dependent interaction used here. The Kuo RPA vector [Kuo] for the 40Ca (3.74 MeV) 3' state is used in this comparison because it is nearly identical to the collective density we have been using for this state. We will also 139 140 examine the effect of using a density which is not the same. For this we will use the Gillet RPA vector [Gil 64] for the 120 (4.44 MeV) 2+ state, which has a different radial dependence than the collective model density. These configurations and the corresponding parameters of the transition density are given in Table 5-1. The densities for 40Ca that will be compared are shown in the tOp half of Figure 5-1. F(r), the radial density defined in equation (3-50), is shown on the left. The solid line is the collective density and the dashed line in the microscopic model given in Table 5-1. On the right side the same conventions are used to show the corresponding electron scattering longitudinal form factor, as defined by equation (E-l6) in Appendix E. It can be seen that the densities are very similar in the surface region, and the corresponding charge form factors are nearly identical over the range of momentum transfer that will be important. The bottom half of Figure 5-1 shows the comparison of the predic- tions of 50 MeV w+ inelastic scattering using these densities. In both cases the solid line shows the collective model calculated using Set D. The left side, labeled (a), shows results that use the four-parameter model with Parameter Set A (dash) and B' (dash-dot) so that the micro- scOpic density is included in a natural way. Set A demonstrates the common deficiency of the IA at this energy (an oddly placed minimum) while Set 3' produces results that are in good agreement with the standard calculation (Set D had a collective form factor) shown with a solid line. That the small differences are due to the use of a dif- ferent interaction can be seen in part (b). Here Set C (dash) and Set D (dash-dot) are used with the microscOpic density and the ansatz 141 Table 5-1 Transition Density for 40Ca and 12C 40Ca RPA Vector and Transition Density Configuration X Y lf7/2 165/2'1 —0.378 -0.201 157/2 281/2‘1 -0.538 -O.236 lf7/2 ld3/2'l -O.736 -0.222 293/2 ld5/2‘1 -0.126 -0.085 2P3/2 ld3/2-1 -O.215 -O.13O 1f5/2 165/2‘1 0.199 0.107 1f5/2 281/2‘1 0.233 0.129 1f5/2 ld3/2‘1 -O.285 -O.163 2P1/2 ld5/2‘1 0.146 0.087 (0.707) a3 [—l.128(ar)3 + 0.9090105] e‘O‘ZrZ a = 0.498 fm‘l 12C RPA Vector and Transition Density Configuration , X Y 191,2 193/2-1 0.91 0.05 1f5/2 193/2-1 —0.08 0.06 117/2 lP3/2'1 0.30 0.02 291/2 lP3/2‘1 0.11 0.08 293/2 1P3,2‘1 —0.12 -0.09 ld3/2 181/2‘1 -0.20 -0.14 135/2 1A1/2'1 0.29 0.20 (0.707) 63 [1.76 (or)2 - 0.057(ar)4] 12'“er 6 = 0.610 fm-l 142 Figure 5-1 Radial transition density (tOp left) and longitudinal form factor (top right) for collective (solid curve) and microscopic (dashed curve) models of the 40Ca (3.74 MeV) 3‘ state, the 50 MeV n+ inelastic scattering calculations at the bottom are described in the text. 143 MSUX-8L 12! 10-2 1 I v T r .C C) ('3 O m 1111:! 0.OL+- « I 0.02~ - F[r‘] I T I 1111] I t I [11"] V 1 1111111 1 11111111 1 1 1 11111 1 -0.02 l L 1 L L L 1 10-50 1\ 3 r [fnfl 10 b r I I I I I I I T I I I I I I T I 10 .- r I I I I I I ff T I I I r Y f: : . : . Z 50 MeV 31 ; q 1 F ‘ 1 .- 1 ._. 1 r -: E : 2 z; : 3 }_ Z J '} _ . .é ' ‘ .§ F ‘ g ‘3 _ 4 \. P '\ b b D u - 0.1,.- 1 0.1: 1 5 I I 1 L' I f . . 1 p 4 r [o] ‘ ' [b] ‘ 1 1 1 1 1 1_1 L L L 1 L L 1 L4 1 1 1 1 1 1 l 1 L 1 0"” 311 6b 911 120 150 0'01 30 so so 120 150 ec.m.[d.9] 9cm.(d°9) Figure 5-1 144 of equation (3-61). The results of the calculations with the same force (Set D) and different densities (collective is the solid curve, micro- scopic is the dash-dot curve) are identical. The result that the effec- tive interaction (8') and the full interaction (D) with this ansatz give comparable results for a microsc0pic density should be expected, and confirms that this method for including second-order effects in micro- scopic calculations is a reasonable one. Additional examples are given in Figure 5-2. The convention at these energies is to use the full potential with theoretical parameters (Set C) for the collective (solid) and microsc0pic (dash-dot) models to indicate the degree of agreement, and the IA (Set A) with the micro- scopic density (dash) to show the result with the first-order potential. The data at 116 MeV and 180 MeV [Mor 80] used in Figures 4-20 and 4-22 are used here. The reader may wish to refer back to these to see the effect of using other forces with the collective model. It is observed that the results of the two calculations differ only at large momentum transfer where the two form factors are different. The IA calculation illustrates the sensitivity to the force. Its predictions are reason- able on resonance, as expected, but less adequate at lower energies and/or large momentum transfer. Overall, we see that the microscopic model gives consistent results if it has the same form as the collective density and if we use the same force in the calculation. The 12C case will show the effects that occur when the form factor has a different shape. The top of Figure 5-3 shows the collective (solid) and microscopic (dashed) form factors as described in Table 5-1. The microscOpic density peaks at a smaller radius, and as a consequence 145 Figure 5-2 Inelastic scattering of N+ and W‘ from the 40Ca 3‘ state at 116 MeV (top row) and 180 MeV, using the microscopic density with sets A and C (dashed and dash-dot curves, respectively) and the collective model with Set C (solid curve). 146 MSUX-8|- |22 qaqq-q d r r I 1 1 I I r T .. u l 1 v I I I -.-» . 4 -ddddd< 4 - .ppppbb b b rpppp . p b “1.5% 30 60 80 120 150 1 E}... as}... 1 o 0 0.01 1.4qqq 1 q :bbppp - p 11qqd q 4 rub-p . p P 90 120 150 ocmld.9) 60 30 1 Egg ou\bo 0.1 1 nw 0 0mm”) fid‘l—W‘d‘ 4 J. ‘dddfl‘ ‘ d d ddd‘l“ 1‘ d1 d1 I I <0. 11 r 1 m I. I I l V A I 10 6 I L I l .1 lo 3 W 1 h-ppp- - - pupppppb b Pup-.pp . p 1 1 m 1 m o. 0 To\nE. ao\bo quqqq a q 1 dqqqqqd q q «<44fidq a 1 f 0 I mu 0 I 5 u: I. I f. o 2 1 I I m T I l o 6 I o 3 r r b-P_. - - pup-ebb. - —--P.- b p , 1 1o 1 .0. o. o. 0 tie... 9S3 amuse) Figure 5-2 147 Figure 5-3 Radial transition density (top left) and longitudinal form factor (top right) for collective (solid curve) and microscopic (dashed curve) models of the 12C (4.44 MeV) 2+ state, the 50 Mev n+ inelastic scattering calculations at the bottom are described in the text. 0.10 0.06 F(r) 0.02 -0.02 10 do/dQ lmb/srl 0.1 0.01 148 10‘1 MSUX-BI-l23 10'2 [F [qu2 10'3 I I [11" I IIIIIII I I IIIIIII I 1 1 1111111 1 111111 1 1111111 1 1 1 1 1 10-... " . T 1 I ' I I 1 1 r v p i I p I! P '1 1- |- ‘ p 50 MeV d b ‘ I- 'l P d p d I- '1 I- d 'l 7 1 . 1 :' 1 .. q ‘_ b 1 u- d I- 1- .1 < I- "‘ - q a I- “ 1- d " é ’ D d ’ d a I- d < I- " :- 1 0.1- ‘1 b d I " p a: 1- " 1- 4 '- d 1- -I 1- " p d I- " p d D -‘ 1- d I- ‘ [a] [b] 1 1 1 IIIVIIVIIUIVIIII lo 1 1 1 1 L 1 , so 60 W 120 150 0'01 OcAIdog) Figure 5-3 30‘ 80 1 1 L L 1 L 90 120 Ounjdog] 150 149 the fourier transform peaks at larger momentum transfer. Some of the consequences of this can be seen at the bottom of Figure 5-3, which use the same conventions as in Figure 5-1. Comparing the collective with Set D [solid curve in both parts (a) and (b)] to the microscopic with the effective interaction Set 3' [dash-dot curve in part (a)] or the full potential Set D [dash-dot curve in part (b)], we see that the shift in the form factor produces an upward shift in the angular dis- tributions. As before, sets 8' and D are roughly equivalent. Two other cases are shown in Figure 5-4. The 68 MeV calculations use Set D for the collective (solid) and microscopic (dash-dot) runs, with the IA (Set A, dashed) also shown for the microscopic form factor. It is extremely interesting that use of this form factor improves the agreement at large angles. The minimum is not well reproduced, but it would appear that these low energy data are sensitive to high momentum components in the wave function. This is due to the p-wave dominance that causes the scattering to be peaked toward 180°. The data at 162 MeV show that this is not the case at higher energies. Here the strong absorption keeps the distorted waves from seeing most of the transition density (which is deeper inside the nuclear surface than for the collective case) and the cross sections are 30-50% lower for the microscopic case. The calculations here used sets D (solid and dash-dot) and A (dash) in the same conventions as before. Thus this density is clearly not a correct description of this state, but it may indicate that the real form factor has high momentum components that are being seen at low energy. 150 Figure 5-4 Inelastic scattering of n+ and n“ from the 12C 2+ state at 68 MeV (tOp row) and 162 MeV, using the microscOpic density with sets A and C (dashed and dash-dot curves, respectively) and the collective model with Set C (solid curve). da/ 69 lmb/ or) do/dQ lmb/erl 10 0.1 0.01 10 0.1 0.01 151 MSUX-Bl-l24 10 I 1 I 11 11111111_L11 30 60 90120150 0cm1dog] Figure 5-4 0.01 l—"- eon.[d°9] 1 1 1 1 l 120 150 1- .- T y 1 I I 1 I r 1 I T I 1 I’I’T I _‘ I- 1 r- / -4 i- ‘1 p. / 1 -1 / .1 .- " / 1- " 1- '4 I- " I- '4 1- J 1- 1 1- c1 - 1 _~ 1 r 1 h d L p '1 '- 4 o 1- 4 i- . p 1 F .1 > p d 1- d g p q 1- " I- '4 a I- " < 1- '1 b u I’ 1 0.1:- 1 I- " n -1 - ‘ P I1 1- " 1. .1 1- “ p- d I- " p d 1- “ r- 4 II- 11+ q p fl- II 1 1 1 1 1 1 1 1 1 1 1 1 1 1 001 1 , 1 1 L L 1 L 1 1 L 1 30 60 90 120 150 ' 30 60 90 150 9cmld°91 omtd.9) . I I r1 I I I I I I I I I I II I 10 1' I y 1 I y T 1 y I ' I I 1 1’4 1- 2 '1 I- j 1- " b n 1 d p I1 _ 162 MeV C q _ . 1- " I- " 1- ‘ I- " .1 I1 : 1 ._ 1r 1 I- '1 g I- d 1- d o p : I- ‘ D _. . ‘E .. . P '1 E I- " i- d a v- " 01 b -1 Q d b u d L- 1 0.1..- q 1- " b " r- 4 I- ‘ r- 4 I- '1 1- .1 1.. c1 1- ‘ h- " 1- 1 . c1 1 152 In summary, it is seen that the ansatz we used for including second order effects in the collective model is also appropriate when microscopic models are used. When the densities are the same, the electron scattering and pion scattering predictions are also the same. Differences in the form factor are reflected in the scattering. The low energy potential is transparent, so the distorted waves can probe the higher momentum components of the nuclear wave function. Combined with the dominance of back angle scattering, this makes the cross sec- tions quite sensitive to these changes. The distorting potential is very absorptive at high energies, effectively cutting off any contri- butions from the parts of the wave function that are inside the strong absorption radius. 5.2 OTHER NATURAL PARITY CASES There will only be a few cases examined here, but they will serve to illustrate some of the interesting problems currently under study. The first cases are states which should be relatively pure shell-model configurations. The 2881 (9.70 MeV) 5' T=O state is predominantly the 1f7/2 d§}2 [01m 79] particle-hole state in a "closed" 2881 core. The 208Pb (6.10 MeV) 12+ neutron state is believed to be a pure excitation of a neutron from the 1113/2 to the 1111/2 shell-model state [Lic 80]. The final case will use the collective model again to examine the effect of the neutron excess of 48Ca on calculations of scattering to the low- lying collective state. The results for the 2381 5‘ state are shown in Figure 5-5. At the top is the electron scattering data from [Yen] compared with the 153 Figure 5-5 Longitudinal form factor (tOp) and 162 MeV w+ and “- inelastic scattering from the 2851 (9.70 MeV) 5‘ state with the two form factors described in the text. 10'2 154 : 1' -3 10 E (\l I 3 i LL P 10‘”: 10'5 1 1 1411111 1 4111111 1 1 144111 O I p... da/dQ lmb/cr) 0.01 I IIIIIII j I IIIIIII I I IIIII I 0.001 ‘ 1 1 1 1 30 80 80 120 OMIdog) '11 TIIIIIIIIIIIIITI 11 J_ 1 150 I l MSUX-BHZS 1 Llllll I ITIT l I 1 A 0.1 L- " g 'r- 4 o u- , § : '4 E 1- on cl z r- b ‘D 1 0.01 :- q r- q I- " b " I- " I- 4 + 11441141111111 _L 1 1111111 1 11111111 1 1111111 1 0.001 Figure 5-5 90 OMIdog] 120 l 1 150 155 longitudinal form factor calculated with a 0.7 (f7/2 dg}2) wave function (dashed curve). The solid curve is the result for a 1.4 (f7/2 d§}2) wave function. This latter case is not unreasonable because a small admixture of the f5/2 dg%2 configuration would produce the same en- hancement. Such a mixture can result from a TDA calculation where the Hamiltonian resulting from the use of the KK [Kal 64] or Elliott [E11 68] force is diagonalized. The densities that result are given in Table 5-2, for comparison with the ones used in these calculations. The pion scattering results at 162 MeV are shown in the bottom half of Figure 5-5 compared to the data of [01m 79]. Both calculations use Set D, the curves correspond to the two form factors shown above. The agreement is quite good -- the strengths of the electron and pion scattering seem to be described in a consistent fashion. The dearth of negative pion data is due to the low beam flux and correspondingly long runs necessary to accumulate good data. The results for the 208Pb 12+ state are shown in Figure 5-6. The electron scattering results indicate that this transition is a pure neutron state [Lic 80] with about 40% of the strength seen. However, since there are no pion data yet available, the form factors here assume 100% of the strength to facilitate scaling the results when the data are analyzed.' Since this is a pure neutron state the charge form factor vanishes; the transverse electric form factor is plotted at the top in the convention given in Appendix E. The pion scattering results use Set D (solid) and Set A (dashed). The two results are quite different because scattering from a neutron is governed by a combination of the T-O and T=1 interactions, and only the T=O interaction includes the 156 Amoco“ bodaamv Numolw mAudv HwH.o d Amom.ov m . I . m em 0 com o :owumfiauamo u=u vwaom Numdlo mAuov mmmo.o mo Amou.ov 0.0 m.o o>u=o cognac Aqmm.o u av hufimcma aofiuwmcmue HIN\mv N\mw HIN\mv «\nm mumum In Hmwm pom moaufimamm aofiufimcmue Nun canoe 157 Figure 5-6 Transverse electric form factor (tOp) and 162 MeV n+ and n” inelastic scattering from the 203m: (6.10 MeV) 12+ pure neutron state using Set A (dashed) and Set D (solid curve) parameters. dcr/dQ (mu/or) 158 MSUX-BI- [26 10‘2 , b T 7 I I d I 208Pb 1 10-3 :- 1 N Z : — - + 1}. ~ 12 v - “T— _ L1. . 4 10‘” .- 1. 10-5 1 1 1 1 - -l 10 ‘ T I I I I I I I I T I r I I’ l I I 4 10 _ I I I I I T I I I I I I T I I I u- “ ' 4 b .4 P 1 I- 1 P I- n p 11 l- 1 I- 1 P 1 D 1 P 1 P 1 10-2 10’2 . 1 b 1 g I- ‘ I- 1 o I- d -1 I- I: 1 > y- -1 b 1 E b d a ‘ P 1 I- C) 4 ‘3 L a F' \ 10 31,- 1 10 : 1 1- : F' .1 I. I. c: ' I - . p 1 I- q r- 4 I- <1 i- 1 TT 1 4L '1‘. 1 1 1 1 1 1 1 1 1 1 L 1 1 1 1° 33 so so 120 150 Omtdog) OcAIdog) 10'“, 30 Figure 5-6 159 second order absorption corrections. The IA calculations give a ratio of 9.2 to l (fl‘/W+) at the first peak, in accord with the expected ratio of 9:1 that would result from the P33 channel alone. The results with Set D only give 3.6 to 1, showing that the results are very sensitive to the density dependent corrections to the isoscalar interaction. Study of reactions involving neutron or T=1 states with normal parity will contribute a great deal to knowledge of the isovector interaction. A related problem is the description of the systematic changes that result as neutrons are added to a T=O nucleus like 40Ca. We have already seen the prediction for 180 MeV scattering from 40Ca (3‘) in Figure 4-22. The t0p half of Figure 5-7 shows the corresponding pre- dictions for the 48Ca (3.83) 2+ state along with the elastic scattering results. The curves use Set D (solid) and Set C (dashed) as in Fig- ure 4-22. The agreement is remarkably good; the shift of the 48Ca n" data towards forward angles is correctly reproduced. The lack of a minimum in the inelastic data is the major defect. Reference to Fig- ure 4-22 shows that the 48Ca data is better explained than the 40Ca data, since there is too much of a shift of the w‘ calculation, espe- cially in the inelastic case. This difficulty is clear in the direct comparison of n+ (solid) and w‘ (dashed) calculations in the bottom half of Figure 5-7, where the excessive shift for 40Ca is easily seen. This problem may originate in the interplay between the coulomb potential and the velocity dependent p-wave interaction (noted earlier for 50 MeV w“ scattering), but it must be understood before conclusions can be reached about the neutron components in these wave functions. 160 Figure 5-7 The top row shows elastic and inelastic scattering of 180 MeV n+ and w‘ from 48Ca and its 3.83 MeV (2+) state with Set C (dashed) and Set D (solid curve), the bottom row compares 180 MeV w+ (solid curve) and n‘ (dashed curve) scattering from 40Ca (3') and 48Ca (2+) with Set D. 161 3. + < 1 1111111 MSUX-81427 1000. 1000. I'IIIIIIIIIIfiIII 100.E a 10.! '5 3 L; g ‘E > 2! = k 2% ; s g E i g 1 - ‘ \ .0 1.! “a g 4.! 1. .01! 001 1 1 1 1 1 1 1 1 1 1 1 1 L 4_JI_J__|,_ ° 30 60 90 120 150 00.1mm.” IOOOOTIIFIIIIIIIIIIIIIV IOOOOIIITIIIIIIIIIIIII 100. 10. da/dQ (mb/or) :5 da/dQ [nob/or] °°°‘ 30 so so 120 ISO Figure 5-7 162 In summary, a few applications of microscopic densities to natural parity transitions have been shown. The 5' case illustrates how this method can be applied, while the 208Pb 12+ shows how one may learn more about the interaction from an investigation of transitions of this type. Results comparing 40Ca and 48Ca states showed the difficulty in inter- preting the origins of differences in the cross sections. 5.3 UNNATURAL PARITY TRANSITIONS States which have a "stretched" configuration have come under a great deal of study lately. These are states for which the particle and hole come from different major shells but have j = l + 1/2, so when coupled to the maximum total J they have J = L+l. These S=l transi- tions can only be produced by the spin-orbit part of the pion-nucleus interaction, and thus test this part of the potential. The 6’ T=l state in 2881 at 14.36 MeV excitation has been very heavily studied with a number of different reactions and thus provides a good testing ground where uncertainties in the wave function can be minimized. First, however, we will give a short review of the method and force used for this calculation. Appendix E outlines the folding model for both central and spin-orbit forces. The essential results for the spin-orbit force are given in equations (3-64), (3-72) and (3-73). The former leads to a recoupling in the distorted waves to reflect the s-1 nature of the transition, while the latter two show how the force parameters found in Chapter 2 are used in obtaining the form factor. Table 5-3 summarizes representative values of the spin-orbit parameters, which were previously plotted in Figure 2-1. 163 Table 5-3 Spin-Orbit Parameters from Rowe, Salomon and Landau 50 MeV 100 MeV 162 MeV 180 MeV so 0.49 0.51 0.24 0.098 +0.044 1 +0.19 1 +0.48 1 +0.47 1 81 0.22 0.25 0.14 0.69 +0.22 1 +0.094 1 +0.24 1 +0.23 1 The 6‘ T=l state in 2881 was first observed with electron scat- tering [Don 70]. Analysis of these data would indicate that 59% of the f7/2 d§}2 configuration was seen. Since this is the only particle-hole combination that can produce J=6 unless one goes up to 3 hu1excitations, it was assumed to be a pure configuration. More recently, it has been seen with inelastic proton scattering [Ada 77] with an angular distribu- tion which supported this spin assignment. New electron scattering data, taken with high resolution, indicate that only 33% of the particle-hole strength is seen [Yen 80]. This will be the figure adopted for this state. The proton scattering data, analyzed in a consistent fashion, suggest only 29% [Pet 80], but there are uncertainties in the choice of the nucleon-nucleon interaction force. Resonant proton scattering from 27A1 give a width for this state which is consistent with a number be- tween these two [Hal 81]. The electron scattering data are shown at the top of Figure 5-8, along with the transverse magnetic form factor calculated from the \/0.33 (f7/2 d§}2) wave function and a a 0.524. The agreement is 164 Figure 5-8 Transverse magnetic form factor (tOp) and 162 MeV n+ and n‘ inelastic scattering from the 2881 (14.36 MeV) 6‘ state with the force and microscOpic form factor described in the text. —~‘_ — “~—_—_ ._-—___—- dCI/dQ lmb/srl 165 10'2 , I . r I 111111 1 IIIIIU] I I IIIUYII I 10's 1 1 288'. 1111 11111111 1 11111111 1 1 10'1 111111 10‘2 U 1 11111] I 10‘3 I IFYIII I V l MSUX-8i-l28 I 1 r- d I- 4 I- .1 n « 10'2— cI a—g I" «1 ‘- _ 4 Q r- d \ 1- cl .0 1— ‘ E I- H '1 7- C) .1 Q l- 8 _ -3 . 10 t' Z Z '1 P d h— ‘ r- d P a! i- 1 1 1 1 1 10'” L 1 1 111111 1 111111 1 1 1 111111 1 1 so‘ 1 1 1 1 90 120 150 9c.m.[d°9] Figure 5-8 90‘ ec.m.[d99] 1 l 1 l 120 150 166 quite good. The bottom half of this figure shows the 162 MeV pion scat- tering data [01m 79] compared to a calculation assuming the same wave function and the force tabulated above. This calculation agrees well with the w+ data, but seems systematically higher than the sparse “- data. The w+ would also seem to prefer a form factor that peaked at higher momentum transfer, but the overall agreement supports the use of this parametrization of the force. In summary, the calculations of pion scattering to an unnatural parity state using the IA value of the parameters, the lab to center- of-mass transformation and the folding model are in agreement with that data when the wave function determined by electron scattering is used. A number of experiments suggest that the wave function is well known, but the absolute uncertainty of 20% in the pion experiments makes a firm conclusion about the spin-orbit interaction impossible. Only the accumulation of more data, some of which is in the next sec- tion, will allow a better understanding of what the effective strength of this interaction must be. 5.4 OTHER UNNATURAL PARITY CASES There are a large number of experiments currently being performed to study other unnatural parity transitions, as well as to study the ones described here at other energies. However, the available data is still quite limited and so only two other cases will be examined here. We will first look at the 12C 1+ T-l state (at 15.11 MeV), and then at the isospin mixed 4‘ states in 16O. The 120 1+ T-l state has been heavily studied because of its impor- tance in predictions of pion condensates and pre-critical opalescence 167 phenomena. Thus there is a large amount of new, accurate data from elec- tron scattering. These data [Che 73, Fla 79, Neuh] are shown in the top half of Figure 5-9. The curve drawn there is a calculation of the trans- verse magnetic form factor calculated from the RPA form factor of Gillet [G11 64] scaled by S2 = 0.27. These parameters are given in Table 5-4. The differences at large momentum transfer are common to all shell model descriptions, and are the focus of the discussions concerning pion con- densation [Tok 80, Del 81]. Since the pion scattering data do not yet reach large momentum transfer, such results do not influence the results we will examine. The results of a calculation of the pion cross section is shown in the bottom half of Figure 5-9. The density used was the same as for the electron scattering calculation, and the pion interaction used was the same as before except that this calculation was for 180 MeV. The Agreement is fair, but the data parameters were given in Table 5-3. [Mor 80] have large uncertainties. The calculation could be low by as Table 5-4 Transition Density for 120 1+ State Configuration X Y 121/2 193/2‘1 1.00 -0.06 1:5/2 123/2‘1 0.02 0.01 2P1/2 123/2‘1 -0.06 -0.01 293/2 193/2-1 -0.06 -0.01 231/2 131/2'1 0.01 0.01 ld3/2 131/2'1 -0.02 ~0.01 Scaled by S = 0.522 (32 - 0.27) 168 Figure 5-9 Transverse magnetic form factor (tOp) and 180 MeV n+ and n' inelastic scattering from the 12C (15.11 MeV) 1+ state, as described in the text. 169 MSUX-Bl-l29 qua—q q 1 qqaaaua « « —q.1<<. . q 3 2C v... I 1 " __ . I TI . on .12 11 .uPF» - “urns-b» . p S_:-._»L . 0 3 _ . . . O 0 0 0 1 1 1 N :32: q [fm'll ».p»-»p » b —dwz ow.oH cam mn.ma .m¢.va 00H mzu aoum mafiuouumom owummaoca I: cam += >0: NoH was Aummav nouumw Show uwumcwma omum>mcmue os-m mtzwsa 172 MSUX-81430 111111 I 1 [111111 1 I [111111 1 r [111111 1 I [111111 I 1 I. . > i (D o - 2 Z 9 b .. a) CD - 0) l\ o o 0 91‘ - m [\ '- v—O '_. - u-o—o ° 0'“ F“ m " an u-o-c .- "°" ’0‘ #0! F." O '- i“ (D D h u- D m h b 11 1 1 111111 1 1 1 1L111111 1 1 1 11111111 1 1 111111 1 1 — '1 ” -: ”-: —' ~ :1 o o q 8 ° :5 [Jo/qw) cp/op rvvv' I t {11111 I v I 'vvrtt 1 I I [11111 T t [tivvrt v o ' D m j- I- F! t ,. H o . II II D D I- h“ ”'0 1.1 u- s... 0.0 I- * * w. 00-. F“ '- v.1 FOO - D b u- < b 11 1 1111111 1 1 111111111 1 1 [11111111 1 1111111 1 ~ o-o ’I—o ’I—n .-. o-I o' d 9' q o o s (JO/mu) tsp/op I'VV'I r ltvvrtr v v 'vvvrrt t v [tvvrlt 1 1 11r1111 v m o ' I’m .4 H ~ “N v- d b “FIG ) Q C) - u ” ‘ r- ’— 1111 1 1 111111 1 1 1 111111 1 L 1 111111 1 1 L 111111 1 1 Q ~ 9 =r '9 ‘9 9 I 1 o O O O 2 2 '0 F4 o-. z 1 [[3le I °¢m(d.9] 06.111.19.91 q Um") Figure 5-10 173 The predictions for the 162 MeV pion data are shown on the right side. The results are consistently 30% low, but the ratios of fi+/fi‘ are in good agreement with the data as shown in Table 5-6. Equally interesting is the prediction of the T=0Tr electron scattering strength, which is quite accurate. The fact that the T=0 data from the pion ex- periment tend to peak at higher momentum transfer may be hinted at in the electron data as well. This may indicate differences in the radial distributions of the neutron and proton components of these wave func- tions (assumed the same here), which could also provide insight into these states. In summary, it is seen that the pion predictions based on existing electron scattering data are in reasonable agreement with the data. These cases suggest that the calculations are 30-50% low, in contrast to the results with the 6‘ state in Section 5.3, but the 202 systematic errors quoted for all of these experiments make conclusions difficult. The shape of the angular distributions are usually correctly reproduced. Table 5-6 Cross-Section Ratios (w+/n‘) for 160 4' States + Tr+ -n: (Expt) -f: (Theory) Energy w W 17.79 1.59 i 0.12 1.68 18.98 0.96 f 0.08 1.00 19.80 0.60 1+ 0.05 0.61 174 5.5 SUMMARY The means to use a microscOpic description of the inelastic scat- tering form factor was introduced. This description was seen to be quite useful when the state being considered could be expressed as a simple shell model state with a spectroscOpic factor determined by other probes. A convenient ansatz, which replaces the collective den- sity with the microscOpic one, allowed the use of the full second order interaction for natural parity transitions. The results were consistent with those of the "equivalent" four-parameter potential, suggesting that this is a reasonable solution to the problem of including density dependence in the inelastic scattering calculation. It was also found that low energy pions are very sensitive to high-momentum components of the density. Thus it may be important to use microscOpic densities, since this allows control of the geometry of the inelastic density independent of the ground state density used for the elastic scattering. Results for transitions with unnatural parity were also presented. The simple IA force described does an adequate job of fitting the well known 6‘ T-l state in 2881. However, it was low for calculations of other states, both of which are also well known from electron scattering. The large systematic errors of these eXperiments limit our conclusions, since it is impossible to distinguish between these calculations and those using the energy shift [Cot 80] which would be 152 higher. Pre- dictions of cross section ratios, which are crucial to the extraction of the isospin mixing coefficients, are not affected by this normaliza- tion problem. These results are within experimental errors, but dif- ferent from what was assumed in the calculation of the wave functions 175 [H01 80], where A33 dominance was assumed. The isospin mixing may need to be reevaluated in this context as better data (especially for the T=0 electron scattering) is obtained. Most of the uncertainties described here must await more data before they can be resolved. A similar situation exists for the T=l part of the interaction for natural parity transitions. The calculation for the neutron state in 208Pb showed great sensitivity to the corrections in the T=O part of the force. Similar corrections in the isovector part should also be easily identified by studying states of this type. The isovector interaction also contributes to differences in scattering from 4003 and 4803. This is difficult to disentangle from the effect of the coulomb force via the velocity dependence, so it's not as useful in determining the force. Pure isovector transitions, and charge exchange, are the best means to identify the prOperties of the effective isovector strength. CHAPTER 6 CHARGE EXCHANGE SCATTERING The charge exchange reaction provides information complimentary to that described so far. This reaction is only sensitive to the isovector part of the force, which has not been studied very extensively. It also provides a consistency check on Optical model predictions of quasi- elastic cross sections. The recent construction of a n° spectrometer at LAMPF has made practical study of these reactions possible. This short chapter will set up the basic foundation for these calculations, and then investigate the prOperties of the reaction for a few cases. Section 6.1 will outline the additional theory necessary for the discussion of single charge exchange (SCE) reactions. Sec- tion 6.2 will then examine some representative calculations as a test of the simple model. 6.1 MODEL FOR CHARGE EXCHANGE CALCULATIONS This section outlines the basic formulae that are needed in addi- tion to the standard ones in Chapter 3. We first work out the operators involved to see how they change for SCE reactions. The form factors are expressed both in terms of the Lane model and a microscOpic density, to allow some of the comparisons we made in Chapter 5. Charge exchange is produced by the isovector part of the potential, which is ‘31 + + 50(1’) ‘3- V’ 59(1') V t'r p b - - 1 1 1 (6-1) a <1 - 11) c1 v2 f 176 177 to first order, neglecting the spin-orbit contribution, as defined in equation (2-43). The operator E'I is of interest because (in the Lane [Lan 62] model) it produces the factors that are different between elastic and SCE reactions. This is t'T a t T + t T + t T (6-2) ~ ~ x x yy 22 which can be rewritten more transparently using raising and lowering operators to give E’T = t T - t T r t T (6-3) and similarly for T. These operators have the property that + 3'" >= 0 T+lp>' 0 + + a = 6-4 tol"> |"> T0|P> Iv) < > + a!“ >- v°> alp>=v2ln> since To a 2 t5 and It =‘\/2 tN. From these definitions it is readily seen that the form of equation (6-3) is consistent with conservation of charge during the reaction. For the purposes of illustration, this operation will be evaluated for the case of 1+ scattering from a target with total isospin T and projection Tz - (Z-N)/2. Recalling that the total isospin operator is 178 just '1‘ =- 21: t: summed over the nucleons in the target, the formula reduces to 5-3 a 2:0 To -\/_2(t+T_ + t:_'r+ . (6-5) For the case mentioned we get t°T|Tr+> l'r'r =2: 1r+ '1' 1'1: ~ ~ ’ Z> o > 0' Z> + 2t+l11 >T_|TTZ> - 2:: 1r+ T TT (6-6) W -| >+| z> .1. 2'11 >TleTz>- 0 -\/3l1r+>-\/T(T+l) - 12024-1) |TTZ+1> which becomes (Z-N)'1r+> In» -\/'2’ N-Z l1r‘> 'TTZ+1> (6-7) for the case where T = -Tz, as will be the situation for the targets studied here. The first term is the contribution to elastic scattering, which enters as s" (z—g’il p(r) (6-8) in equation (2-44). The factor of A is required to convert p(r) to a single nucleon density. The second term is the contribution to SCE, and can be parametrized in two ways. 179 The Lane model [Lan 62] is a straightforward extension of equa- tion (6-7) to give 2 ”‘2 p(r) <6-9) A for the transition density in SCE. By using p(r)/A, this essentially assumes that all the nucleons participate equally, ignoring any speci- fics of nuclear structure. This is the same assumption made in writing equation (6-8), but it is far more important in equation (6-9) than in a small correction term like equation (6-8). The equivalent micro- scOpic density is found by replacing equation (6-9) by the overlap Of a specific final state with\/§ T- Operating on the ground state. For an analog transition like 13C + 13N, this becomes just VS pmsm 100m (6-10) where DIAS is the density of the specific orbital involved (the 91/2 in the case cited). As a practical note, the algebra in DWPI assumes that a YLM explicitly multiplies the form factor so that a correct normalization of equation (6-9) requires the addition ofurz; YOOr while this Operator automatically appears from the formulae used to obtain equation (6-10). 6.2 SAMPLE CALCULATIONS The calculations shown here will all employ the IA form of the potential in equation (6-1), primarily because the information about the isovector part of the interaction is limited. In this way the results will be amenable to scaling and the determination Of the ef- fective strength of the isovector potential. The parameters used will be those already defined for Set A in other parts of this thesis. 180 The first case to be studied is the reaction 13C (N+,W°) 13N. This has a long history, with total cross sections first measured using nuclear chemistry [Chi 69, Zai 73, Sha 76]. These data have proven very difficult to explain, typical results can be found in [Gib 76, Spa 79]. More recent results with the fl° spectrometer at LAMPF have provided angular distributions and a confirmation of the Older total cross section measurements [Dor 79]. The results of the Lane model calculations are shown with a dashed line in Figures 6-1 and 6-2. The 162 MeV calculation is similar in shape but much lower than the 150 MeV (solid points) and 180 MeV (Open points) data, as seen on the right side of Figure 6-1. Figure 6-2 shows that these calculations have the same steep falloff in the resonance region as is seen in other calcu- lations. The data shown are [Zai 73] and [Sha 76]. The solid line in these figures shows the result with the micro- scOpic description in terms Of a pure pl/2 particle involved in the analog transition. The results are quite different, particularly the low energy angular distribution, and the cause is not immediately evi- dent. What is interesting is that the cross sections do not fall off rapidly with the increase in bombarding energy. Some of this (about 202 at high energy) is attributable to the inclusion Of spin-flip in these calculations, but most of the difference must be due to the onset of strong absorption near 100 MeV. We have already seen that the dis- tortion in this region keeps the pion from seeing the interior wave function. The Lane density is mostly inside the strong absorption radius, so these cross sections are reduced by a large factor. 181 .Am>u:o wHHomv Hovoa oaowuuma mamcwm was Am>u=o wmnmmvv Hmwoa mama onu magma .UMH :O.+= >mz NoH mam >oz on nuas mwcmnoxm owumno mecfim mo :Owusnauumfiv umaawc< Hue whawfie MSUX-8l-l3l 182 111111 1 1 [1111111 1 [ITFITT r I 111111 1 1 I / > / - CD I , Z \ \\ (\l \\ o — (.0 \\ '13- !—'1 V l’l” u- ’I’ .1 _ _.O (V 111111 1 1 1111111 1 L 1111111 1 1 1111111 1 1 .—1 v-O v-1 9 " c3 :5 8. O [.K?//QLU] ESF%//I3p 1111111 T [111111 1 1 1111111 1 1 [111111 1 1 \ > \ (D \ - z I - 1 O I, ‘0 r If) I! 3- / / 1- / q / I, i— ‘0 I, N I, 1- q 1 1 11111 1 1111LL1 1 1 1111111 1 1 111111 LL 1 o-1 H c-1 9 " o' 2 a: o [Js/qw] esp/op ecm [deg] ecm [deg] Figure 6-1 183 .mawvoe Am>u=o vuaomv oaoauuma mawcfim was Avmsmmuv mama mafia: .omH co mwcmnoxo mwumso mawcam += u0m coauocam :Ofiumuaoxm «no shaman MSUX-Bl-BZ 184 11111 [111111 1 1 [IIIIIFI 1 1+4 1 I PM 1 I O l N 1 I I I I O _ "C3 H D d P—O—-—1 111111 L 11111111 1 1111111 1 v-1 0 o O [qw] D E (MeV) Figure 6-2 185 Of similar interest is the mass dependence of the cross sections at a fixed energy. These were all calculated in the Lane model to test the consistency of this with another simple model of SCE reactions [Joh 79]. The results are shown in Figure 6-3, scaled to the 208Pb point. These values were all given in relative cross section, the calculated values for the 13C and 208Pb cross sections at 5° were 0.22 mb and 0.18 mb, respectively. At this energy the angular distri- butions of different models are similar, so this should be a reasonable prediction apart from the overall scale. These agree fairly well with the data, and are quite similar to the calculations of Mikkel Johnson's [Joh 79] semi-classical model. Finally, the Lane (dashed) and microscOpic (solid) results are plotted for the 15M (1+,n°) 150 case at 50 MeV as shown in Figure 6-4. The predicted angular distributions here are very different. The Lane results are consistent with those for 13C, but the other calculation has drOpped significantly. The situation here is confusing, and may reflect some of the sensitivity to the distorted waves and choice of transition density that we saw in other low energy data. 6.3 SUMMARY The method for doing a simple calculation of pion single charge exchange was outlined. The results are in rough agreement with low energy cross sections, but are systematically low at higher energies. Whether this is due to the need to use a stronger interaction or a more realistic form factor needs to be studied in parallel with other isovector transitions. 186 .Hmuos mama wnu magma mama umwumu :o mmcmzoxo mwumso oawcfim +e >mz OOH mo moaocamamo muo magmas 187 owmw mum spawns < 2: mm_-_m-xam_>_ [shun 'qqo] [cg] asp/op 188 Figure 6-4 Single charge exchange with 50 MeV 1+ on 15N, calculated with the Lane (dashed) and single particle (solid curve) model. dcr/dQ [mb/sr‘] 10 0.1 0.01 0.001 189 MSUX-8l-l34 l I - - d 1- d — d — 50 MeV 4 - q I— an I— - _ d — c-l :— d I- .1 I— .- 1— ‘ p d i— 4 c1 P ~‘~ \_ \ .d — \ ’ : ~~__-” : I- c- p— q — 4 -' .1 .J I- - — 1 — u .- ' I i _ d 1- L - I— -I 1 l 1 l 1 220 Gem [deg] Figure 6-4 l+0 CHAPTER 7 CONCLUSIONS In the first half of this work we saw that the impulse approxima- tion and multiple scattering theory could be used to construct a pion~ nucleus interaction that, with some minor adjustments in a few param- eters, accurately describes elastic scattering from very low energies up to 200 MeV. This complicated density dependent interaction was found to be roughly equivalent to a simple four parameter effective interaction, which simplified the discussion of the essential physics introduced by the corrections to the impulse approximation. The theo- retical pion-nucleus isoscalar central interaction, calibrated by the large amount of elastic scattering data, was then tested against iso- scalar natural parity inelastic transitions. The low energy elastic scattering data clearly prefer an effec- tive interaction with increased s-wave repulsion and weakened p-wave attraction. When this interaction is used for inelastic scattering calculations, the collective states are correctly described. The inelastic scattering results are found to be very sensitive to the choice of Optical potential, primarily because the potential is rela- tively transparent at these energies. The absorption required at 50 MeV is less than the value determined by pionic atoms, requiring some further theoretical study. The resonance region data are reasonably described by the theore- tical parameters. The elastic cross sections are not affected much by changes in the interaction, except at back angles where there is little data. Again it is found that the isoscalar interaction preferred by 190 191 the elastic data is also preferred by the inelastic scattering data. Here, however, the inelastic calculations are quite insensitive to the choice of distorting potential (mainly because the potential is strongly absorbing in each case), so it is more productive to investigate prop- erties of the force by looking at inelastic scattering reactions. Reactions that investigate the transition region (60-100 MeV) seem complicated and are difficult to interpret. Part of this is due to the fact that diffraction effects coexist with effects associated with properties of the interaction, making it hard to disentangle the cause responsible for the observed angular distributions. There are some interesting cases that seem to suggest that the pion may be a good probe of the high momentum components of the wave function at these beam energies. ' In summary, we saw in Chapter 4 that the isoscalar central inter- ation, obtained from theory and calibrated with elastic scattering data, gave a good description of collective inelastic transitions. Further study of the Optical potential and inelastic scattering to various natural parity states should help clarify the details that are still missing. Clearly the ultimate test of the theory is to produce a de- scription of the data at 80 MeV that is consistent with the high and low energy results. There is not very much information with which to test the iso- vector central interaction. The evidence from neutron states is that the isovector part of the force is reasonable, although it is difficult to separate it from the effects of the coulomb force on the velocity 192 dependent p-wave interaction. Charge exchange reactions provide com- plimentary information on the isovector interaction, but seem to imply that the defect is mainly in our understanding of the reaction rather than in any particular part of the force. Further work is needed, particularly in obtaining a better understanding of the medium correc- tions to the isoscalar central interaction. Information about the spin-orbit interaction can be obtained only from inelastic transitions to unnatural parity states. The evidence from such spin-flip transitions is that the spin-orbit interaction is reasonably described by the impulse approximation, but may need to be increased by lS-ZOZ at 165 MeV. No information exists for these reac- tions at other energies. The simple structure of the stretched states has made them ideal places to test the interaction, and a systematic collection of data should contribute a great deal to our understanding of this part of the pion-nucleon interaction. In summary, much is understood but much more remains to be studied. In particular, the steady improvements in beam quality and detector resolution will increase the amount of data in areas that have currently only been lightly surveyed due to limitations of the experimental equip- ment. One purpose of this work has been to identify consistencies, as in the isoscalar interaction, and point towards areas of interest, such as the spinrflip transitions and studies of the proton and neutron components of a wave function, in order to contribute to this process of choosing good places to test the theory and study nuclear physics. Identification of the prOperties of the effective interaction and its application to the consistent interpretation of a large set of data is one means to this end. APPENDICES APPENDIX A PION SCATTERING AMPLITUDE This appendix gives the formal conventions used in the definition of the pion scattering parameters in terms of the pion-nucleon phase shifts. The derivation essentially follows that of [Str 79a, E13 80]. The scattering amplitude is expanded in terms of projection operators for the total isospin T, and the angular momentum L and total angular momentum J - L t 1/2. The result is L a .- f E PT PM aw,” (2L+1) PL (cos 9) (A 1) 'r L,J L 216 L a 2T,2J_ where OZT,2J (e €L/éik is defined by the phase shifts and the projection operators are 1 1 P1/2 ' 3' (1 ' 5'5) P3/2 3 (2 + 5'3) -> + -> + (A-Z) PLJ< a (L " (I'M/(2L+1) PM) ’ (L+1 4' CI'M/(2L+1) . We are limiting the discussion to s- and p-wave cases, so the sum on L,J gives 1» ° + 1 (1 +71: 1 (2+7? 9 - -— o + o f 2 1‘: 51,1] 0‘21,1 ° ) 021,3 0 ) °°8 1' + since £Po(cos 6) vanishes. We can rewrite this as o 1 1 a '1' f 2;, P11 [“2131] + [“214 2“2133] °°8 e 1 +1 “*7; e + -02T,1 aZT,3 0 cos (Ar3) (Ar4) 193 194 and then do the sum on T. This gives 0 _ . 0 . al,l (l E I)/3 + a3’l (2 + E :)/3 (ArS) a; o 0 [( al,l+2a3,l + O O _. + 0 0‘1,1 a3,l) E I ] for the s-wave part, and similar results for the others. Combined, these give -a1 + a - 2a ’3 + 2a:,3) 2': ]k2 cos 9 (A-6) ++ Since k2 cos 6 - k‘k', and + + «a i cos 9 = -i r x e 36- cos 8 (Ar7) +-> + + + gives k2 0'1 cos 9 = 10 ° (k x k'), equation (Ar6) defines b0, b1, co, c1, so, 31 in equation (2-26). Finally, we note that the values of the GET 2J are obtained from D the phase shifts GET 2 as parametrized by Rowe, Salomon and Landau [Row 78]. APPENDIX B EQUIVALENT FORMS OF PION OPTICAL POTENTIAL There are two points to be made here. One is to convert the standard Kisslinger potential into a local (Laplacian) form that is suitable for plotting in momentum space. The other is to sketch the conversion of the potential into coordinate space. The local form is obtained by using that q2 -- (IE—12') - (II-P) ++ = k2+ k'z - Zk-k' (B-1) 2 +4» = 2k - 2k'k' , so we use the conversion + + k-k' - k2 - § qz (Is-2) to eliminate EEE' from the expressions for the potential. In order to plot the full potential in equation (2-55) it is necessary to evaluate the additional density-dependent parts of the interaction. This is done following the idea in Section 2.6, where a constant value of o = 0.12 fm‘3 is used. Writing this as peff we then have 1': ‘4“ - (91 b0 + p2 Bo peff) -1 -1 1’1 co + p2 Co peff (k2 _ 1 q2) + 4n -1 -1 1+3_'A[p1 °o+pz Co peff] peff " [(1 ‘ P11) co + (1 ' 1’21) Co peff]-(213 : (B-3) for the t-matrix plotted in Chapter 4. 195 196 ++ The coordinate space versions of k°k' and q2 come from their fourier transforms. The first is 1 11? +' + + I I 3 —-7 e r k'k' p(q) eik 1‘ d3k d k' (2“) (B-4) 1 413-15 + + 11?? 3 3 = 6 e V'°V p(q) e d k d k. (2W) which simplifies if it is rewritten in terms of k+k' q = k-k' . Q "'§‘- . i x - r-r' , X a rgr to give 4- -> . . 1 6 v'-v p(q) eiq x eiQ x d3q d3Q (2N) (B-S) = 3“ (50:) pm 31 + where we have been careful to keep the V Operators acting on the wave functions as defined in equation (B-fi). Using the delta function + reduces this to a function of r, and the V can be reversed to give -;° [p(r) 3] (3‘6) which is the form used here. 197 The q2 term is done in a similar fashion, where it is convenient to start with the variables of equation (B-S) giving q'X eiQ‘x d3q 3 1 6 qz p(Q) 31 d Q (2") 3 1 (‘V2) p(q) eiq'x em.x d3q d3Q (3‘7) (2")6 X 2 a " X p(X) (3(X) 2 which can be reduced to 2 'V p(r) (Ii-8) by using the delta function. APPENDIX C DENSITY PARAMETERS The radial densities used for the nuclear ground states in the Optical potential are in one of two forms. One is the gaussian form g)2 -2 w e with (C-1) -1 p0 - 2[(2 + 3a)(./-1FW)3] , p(r) I p [I + a o while the other is the three-parameter Fermi (or Woods-Saxon) form r 2 1+W(:) p(r) - o r- 1 + exp(‘1E-) with (C-2) -1 2 2 p . 33 1+(12)+§(3+1o[2£]+7[1£]) . o 4" c c where we use a - (4 ln3)t as our input. The parameters are taken from electron scattering results [DeJ 74] with the finite size of the proton removed using R: - R: + 0.64 to convert the charge radii to nuclear radii. The values used are tabu- lated in Table C-l. For convenience, this table also lists the energy and collective model deformation parameter BL [as defined in equa- tion (3-44)] of the collective states studied here. 198 199 TABLE C-l Density Parameters Gaussian Density [Equation (C-l)] Excited States Nucleus wc W o J“ E 8J 120 1.66 1.57 1.33 2+ 4.44 0.60 3‘ 9.63 0.44 160 1.83 1.75 1.54 Three-Parameter Fermi [Equation (C-2)] Excited States .Esslssa cc c a w J" E 53 2831 2.93 2.82 2.50 0 2+ 1.77 0.40 ' 400a 3.67 3.58 2.56 -0.10 3- 3.74 0.39 430a 3.74 3.65 2.30 -0.03 2+ 3.83 0.17 902: 4.83 4.76 2.18 0 3- 2.75 0.14 203Pb 6.46 6.41 2.38 0 3- 2.62 0.12 APPENDIX D ++ EVALUATE V'V IN FORM FACTOR In equation (3-36) it was assumed that the angular integral could ++ be done separately. Since the k-k' part of the interaction looks like {7" [F(r) YLM] E7 , (13-1) _) there is an explicit angular dependence in the v operators that could change the coupling to the angular parts of the wave functions. This is not the case, as was shown by [Edw 71], but the result is important enough to include here. The integral we need to evaluate looks like .[63.34_ Y + F ) 6.32;-Y D 2 r r 1m V [ L -+ -/dr A v° [B v C] (D‘3) for convenience. The gradient can be reversed by integrating by parts, giving 3 -)- + - dr (VA) B (W) (D-4) which can be written as gfdfi [(va)Bc - A(v28)c + AB(VZC)] . (ID-5) 200 201 The proof of the identity used to construct equation (D-S) follows. It is based on combining 6V [0 3 v] = U(Vzv) + ($8) - (EFV) (D-6) and [W (7*. [U E7 v] a - (3W) U(vV) (D-7) to get -f($w> U(§V) -f(V2V) WU =f(3U) W(;V) . (D-8) Reversing the role of U and V gives {who (30) —fvmv20) = (30) w<€v> <0-9) which, when combined with equation (D-8) gives [(30) 1467M = - % U230) (3am +fU<3m (3V) +f(V2U) WV +fUW(V2V)] . If this is written out for each of the three possible combinations (D-lO) of A, B, C, then two can be substituted into the equation for the other giving flgA) ROTC) - - % [%f(V2A)BC -fA(VZB)C + §fAB(v20) -f(§7A)8 (30) .. %fA(;B) (3c) --:- (3A) (gmcj . (D-ll) 202 If we now use the original equation (D-lO) written as - §f<3A> (38m - -:—[A(;B) (30) (D-12) + -> =«fwm B(VC) + %[(V2A)BC + -:-[AB(VZC) we can reduce equation (D-ll) to + + [074) B(VC) .. %[A(VZB)C - -:-f(V2A)BC - i—fAszc) (0-13) which proves the result in equation (D-S). The final step is to separate variables, using v2 aLL rz-d— -L—2 (D‘14) 2 dr dr 2 ’ r r to evaluate the integrals in equation (D-S). As a specific example, examine just one of the terms 2 [(v2 A)BC - 1:5%—(r2 ) BC rzdr an - Lg BC 83 (0-15) where the second term gives 2(£+l) and the first can be integrated by parts to give dC ABC 3 drc+8dr)d§_1(1+1) rzdr . (D16) 28(92. dr If this and the other two corresponding equations are used in equa- tion (D-S), a large amount of cancellation occurs so that we get A v- (B v C) dr -- L(L+1) - 2'(2'+1) - 2(2+1)J f—3 dr (0-17) - {83(1) 8 203 If we now return to the original definition of the terms, we see that we have * -£ [L(L+l) - £'(£'+l) - £(£+l) -:£ F -:£L dr Y Y Y d9 2 r L r 1M LM i'm' (D-l8) * x du u du u _ 2_ 2 2'_ 2' (dr r) FL ( dr r ) dr Yam YLM Yg'm' d9 so that the angular matrix element comes through as assumed in equa- tion (3-36). The object called FL here is the part of AtTTN FA associ- ated with the 3'; term. This would be the A2 F(r) in equation (3-59) or the corresponding terms in equation (3-61). The radial part of equation (D-18) is the factor evaluated by DWPI when constructing the transition matrix element. APPENDIX E FOLDING MODEL FORMULAE This appendix will outline the derivation of the folding model formulae that were given in Chapter 3. In addition, it will define the conventions for electron scattering form factors. The discussion here is for pions only, so the interaction is limited to the spin-orbit and central terms only. It is written as ”/"pt = 8C + game) 1 651* . 02-1) The central part contains the kinematic factors [see equation (2-43)] for the transformation to the pion-nucleus center of mass and will eventually be expressed as a function of q in the local Laplacian model as described in Appendix B. This Laplacian form of the central inter- action is never used for calculations here, but is included to indicate the way the folding formula works. The spinrorbit part is also affected by the kinematical transform but, as shown in equation (2-42) the factor of P11 cancels the factor of p1 from Y [equation (2-33)]. Thus, the spin-orbit comes in without any extra factors. The second term in equa- tion (2-43), which contributes to elastic scattering, would come in with an extra factor of 8. Although a number of terms from the spin-orbit interaction can contribute to various ractions, the discussion here will lbe limited to those that contribute to abnormal parity transitions. One result will be used repeatedly in these calculations. It relates the expansion of a tensor product, which usually looks like Y ° T . The expansion of the Y gives L L L 204 205 L “L 'L Y(f)=(4n)1/ZE 19‘ LLL1 LM 9 t L Lp t (E-Z) L x [YLP ® YLt] M and the rearrangement of the tensor product gives L -L -L -1/2 p t A A A-l ° T - 4n 1 L L YL L ( ) E E Lp t J Lth +L L x (-) P W(Lth klkz; LJ) J J x [pr a k1] - [1% a k2] (2-3) where k1 and k2 stand for the arguments out of which the tensor TL is composed. The central term is straightforward, since C C a 800-“) “44'" g (r) Y00(r) L 'L g (r) Z Z 1 Lp Lt LpL J LP x (-) w(Lth 00; 0J) J J x [YLP e 1] . [YLt ® 1] . (pg-4) 206 Evaluating this, the clebsch gives Lp = Lt and the Racah gives Lp = Lt = J and this all reduces to J go“): [YJQ‘DI]J . [YJ®1] (E-S) J The fourier transform of this gives 2 C 11‘ . .1“ _ an g (k) Z 4163‘?) 4mm (E 6) n J where w2 = 2k2 dk/n , n C 8 (k) = 4fl'Jf j0(kr) gc(r) r2 dr 0 and jJOJ = J'J(kr) YJ(r) . When the expectation value over the target and projectile wave func- tions is taken, J <17JOJ(C)> —.J'J(kr) o (k) and (E-7) <%OJ(9)> - TJOJ(p) where pJ(k) is the fourier transform of the form factor OJ(r) defined in Chapter 3, and TJOJ is the operator used in the angular momentum algebra of Chapter 3. 207 The spin-orbit term requires rewriting ., +++ 81f ,, ++ L ' r°(p x o) = -i '3- r Yl(r)-Tl(p, o) (E-8) using results from Brink and Satcher [Bri 71]. When combined with the original formula in equation (E-l), this gives L3 8 a rg 11(3. 3) L-L-laaa_ stigma/3:216: Lthll L Lt J Lp+l x (-) W(Lth 11, 1.1) x YL ®P YL @o . (E-9) p t The fourier transform of this gives w2 gLs(k)ZZ:I/2 JUL-lJLll (E-lO) J+l , J- x (JO LOIIO> (-) wu L 11, 1.1) x .4JCp) °./L1J(t) where LSm - 4u- 31(82) kr gLS(r) r2 dr (2-11) and e% a l- 1]. + J JJ k J3 x P] (E-lZ) ,~ - J + J ‘/L1J [e/L x o] and git a jL(kr) YL(r) . 208 The other terms, principally the one involving JlJ: have been omitted since they do not contribute to an abnormal parity transition. If we now evaluate the expression, we get an w2 gLS(k) Z 4J9) . 1'1 J__+_1 J 2J+l JJ J-1,1J ,/2J+1'/J+1,1J When the expectation value is taken this becomes 2 LS Z wn g (M Z (‘31) - 0:00 chm (Ia—14) J n (E-13) where jJ is the spherical Bessel function, 3 J+l _ J j- _ J ,/ 2J+l ‘/ 2J+l <'/J+1,1J> (E 15) 1‘ = ‘ (kr) T r2 d <'/ L1J> JL < L1J> r and where is worked out in Chapter 3. The form factor is the and "spin density" for this transition, described in Chapter 3. Because of the form of this result, it is only necessary to calculate one term in DWPI when including the spin-orbit calculation. Finally, it is necessary to summarize the definitions used for the electron scattering results. The charge form factor is 4 2 lFIZ a w 1' +1 ”J( )l (E-l6) ___2 z2 ZI+l 209 where p is the expectation value of <‘jjoj> as defined above. The J transverse electric form factor is 2 2 2 21'+l 2 eh S S L =——- _ _— + - FTE 21+1 (2mc) 2 0JJ 2gL DJ (E 17) where gS and gL are the gyromagnetic ratios for spin and orbital cur- rents; the current densities are defined in [Pet 81]. The magnetic form factor is gs s L 2 2 _ 21'+1 2 ‘__ _ 2 2 DJ gL pJJ ’ 21+1 F M (E-l8) eh 2 2mc where p: is the spin density that enters pion scattering, and the orbital current is defined in [Pet 81]. 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